Modules / Lectures

Module Name | Download |
---|

Sl.No | Chapter Name | MP4 Download |
---|---|---|

1 | Motivation for Counting | Download |

2 | Paper Folding Example | Download |

3 | Rubik's Cube Example | Download |

4 | Factorial Example | Download |

5 | Counting in Computer Science | Download |

6 | Motivation for Catalan numbers | Download |

7 | Rule of Sum and Rule of Product | Download |

8 | Problems on Rule of Sum and Rule of Product | Download |

9 | Factorial Explained | Download |

10 | Proof of n! - Part 1 | Download |

11 | Proof of n! - Part 2 | Download |

12 | Astronomical Numbers | Download |

13 | Permutations - Part 1 | Download |

14 | Permutations - Part 2 | Download |

15 | Permutations - Part 3 | Download |

16 | Permutations - Part 4 | Download |

17 | Problems on Permutations | Download |

18 | Combinations - Part 1 | Download |

19 | Combinations - Part 2 | Download |

20 | Combinations - Part 3 | Download |

21 | Combinations - Part 4 | Download |

22 | Problems on Combinations | Download |

23 | Difference between Permuations and Combinations | Download |

24 | Combination with Repetition - Part 1 | Download |

25 | Combination with Repetition - Part 2 | Download |

26 | Combination with Repetition - Problems | Download |

27 | Binomial theorem | Download |

28 | Applications of Binomial theorem | Download |

29 | Properties of Binomial theorem | Download |

30 | Multinomial theorem | Download |

31 | Problems on Binomial theorem | Download |

32 | Pascal's Triangle | Download |

33 | Fun facts on Pascal's Triangle | Download |

34 | Catalan Numbers - Part 1 | Download |

35 | Catalan Numbers - Part 2 | Download |

36 | Catalan Numbers - Part 3 | Download |

37 | Catalan Numbers - Part 4 | Download |

38 | Examples of Catalan numbers | Download |

39 | Chapter Summary | Download |

40 | Introduction to Set Theory | Download |

41 | Example, definiton and notation | Download |

42 | Sets - Problems Part 1 | Download |

43 | Subsets - Part 1 | Download |

44 | Subsets - Part 2 | Download |

45 | Subsets - Part 3 | Download |

46 | Union and intersections of sets | Download |

47 | Union and intersections of sets - Part 1 | Download |

48 | Union and intersections of sets - Part 2 | Download |

49 | Union and intersections of sets - Part 3 | Download |

50 | Cardinality of Union of two sets - Part 1 | Download |

51 | Cardinality of Union of sets - Part 2 | Download |

52 | Cardinality of Union of three sets | Download |

53 | Power Set - Part 1 | Download |

54 | Power set - Part 2 | Download |

55 | Power set - Part 3 | Download |

56 | Connection betwenn Binomial Theorem and Power Sets | Download |

57 | Power set - Problems | Download |

58 | Complement of a set | Download |

59 | De Morgan's Laws - Part 1 | Download |

60 | De Morgan's Laws - Part 2 | Download |

61 | A proof technique | Download |

62 | De Morgan's Laws - Part 3 | Download |

63 | De Morgan's Laws - Part 4 | Download |

64 | Set difference - Part 1 | Download |

65 | Set difference - Part 2 | Download |

66 | Symmetric difference | Download |

67 | History | Download |

68 | Summary | Download |

69 | Motivational example | Download |

70 | Introduction to Statements | Download |

71 | Examples and Non-examples of Statements | Download |

72 | Introduction to Negation | Download |

73 | Negation - Explanation | Download |

74 | Negation - Truthtable | Download |

75 | Examples for Negation | Download |

76 | Motivation for OR operator | Download |

77 | Introduction to OR operator | Download |

78 | Truthtable for OR operator | Download |

79 | OR operator for 3 Variables | Download |

80 | Truthtable for AND operator | Download |

81 | AND operator for 3 Variables | Download |

82 | Primitive and Compound statements - Part 1 | Download |

83 | Primitive and Compound statements - Part 2 | Download |

84 | Problems involoving NOT, OR and AND operators | Download |

85 | Introduction to implication | Download |

86 | Examples and Non-examples of Implication - Part 1 | Download |

87 | Examples and Non-examples of Implication - Part 2 | Download |

88 | Explanation of Implication | Download |

89 | Introduction to Double Implication | Download |

90 | Explanation of Double Implication | Download |

91 | Converse, Inverse and Contrapositive | Download |

92 | XOR operator - Part 1 | Download |

93 | XOR operator - Part 2 | Download |

94 | XOR operator - Part 3 | Download |

95 | Problems | Download |

96 | Tautology, Contradiction - Part 1 | Download |

97 | Tautology, Contradiction - Part 2 | Download |

98 | Tautology, Contradiction - Part 3 | Download |

99 | SAT Problem - Part 1 | Download |

100 | SAT Problem - Part 2 | Download |

101 | Logical Equivalence - Part 1 | Download |

102 | Logical Equivalence - Part 2 | Download |

103 | Logical Equivalence - Part 3 | Download |

104 | Logical Equivalence - Part 4 | Download |

105 | Motivation for laws of logic | Download |

106 | Double negation - Part 1 | Download |

107 | Double negation - Part 2 | Download |

108 | Laws of Logic | Download |

109 | De Morgan's Law - Part 1 | Download |

110 | De Morgan's Law - Part 2 | Download |

111 | Rules of Inferences - Part 1 | Download |

112 | Rules of Inferences - Part 2 | Download |

113 | Rules of Inferences - Part 3 | Download |

114 | Rules of Inferences - Part 4 | Download |

115 | Rules of Inferences - Part 5 | Download |

116 | Rules of Inferences - Part 6 | Download |

117 | Rules of Inferences - Part 7 | Download |

118 | Conclusion | Download |

119 | Introduction to Relation | Download |

120 | Graphical Representation of a Relation | Download |

121 | Various sets | Download |

122 | Matrix Representation of a Relation | Download |

123 | Relation - An Example | Download |

124 | Cartesian Product | Download |

125 | Set Representation of a Relation | Download |

126 | Revisiting Representations of a Relation | Download |

127 | Examples of Relations | Download |

128 | Number of relations - Part 1 | Download |

129 | Number of relations - Part 2 | Download |

130 | Reflexive relation - Introduction | Download |

131 | Example of a Reflexive relation | Download |

132 | Reflexive relation - Matrix representation | Download |

133 | Number of Reflexive relations | Download |

134 | Symmetric Relation - Introduction | Download |

135 | Symmetric Relation - Matrix representation | Download |

136 | Symmetric Relation - Examples and non examples | Download |

137 | Parallel lines revisited | Download |

138 | Number of symmetric relations - Part 1 | Download |

139 | Number of symmetric relations - Part 2 | Download |

140 | Examples of Reflexive and Symmetric Relations | Download |

141 | Pattern | Download |

142 | Transitive relation - Examples and non examples | Download |

143 | Antisymmetric relation | Download |

144 | Examples of Transitive and Antisymmetric Relation | Download |

145 | Antisymmetric - Graphical representation | Download |

146 | Antisymmetric - Matrix representation | Download |

147 | Number of Antisymmetric relations | Download |

148 | Condition for relation to be reflexive | Download |

149 | Few notations | Download |

150 | Condition for relation to be reflexive. | Download |

151 | Condition for relation to be reflexive.. | Download |

152 | Condition for relation to be symmetric | Download |

153 | Condition for relation to be symmetric. | Download |

154 | Condition for relation to be antisymmetric | Download |

155 | Equivalence relation | Download |

156 | Equivalence relation - Example 4 | Download |

157 | Partition - Part 1 | Download |

158 | Partition - Part 2 | Download |

159 | Partition - Part 3 | Download |

160 | Partition - Part 4 | Download |

161 | Partition - Part 5 | Download |

162 | Partition - Part 5. | Download |

163 | Motivational Example - 1 | Download |

164 | Motivational Example - 2 | Download |

165 | Commonality in examples | Download |

166 | Motivational Example - 3 | Download |

167 | Example - 4 Explanation | Download |

168 | Introduction to functions | Download |

169 | Defintion of a function - Part 1 | Download |

170 | Defintion of a function - Part 2 | Download |

171 | Defintion of a function - Part 3 | Download |

172 | Relations vs Functions - Part 1 | Download |

173 | Relations vs Functions - Part 2 | Download |

174 | Introduction to One-One Function | Download |

175 | One-One Function - Example 1 | Download |

176 | One-One Function - Example 2 | Download |

177 | One-One Function - Example 3 | Download |

178 | Proving a Function is One-One | Download |

179 | Examples and Non- examples of One-One function | Download |

180 | Cardinality condition in One-One function - Part 1 | Download |

181 | Cardinality condition in One-One function - Part 2 | Download |

182 | Introduction to Onto Function - Part 1 | Download |

183 | Introduction to Onto Function - Part 2 | Download |

184 | Definition of Onto Function | Download |

185 | Examples of Onto Function | Download |

186 | Cardinality condition in Onto function - Part 1 | Download |

187 | Cardinality condition in Onto function - Part 2 | Download |

188 | Introduction to Bijection | Download |

189 | Examples of Bijection | Download |

190 | Cardinality condition in Bijection - Part 1 | Download |

191 | Cardinality condition in Bijection - Part 2 | Download |

192 | Counting number of functions | Download |

193 | Number of functions | Download |

194 | Number of One-One functions - Part 1 | Download |

195 | Number of One-One functions - Part 2 | Download |

196 | Number of One-One functions - Part 3 | Download |

197 | Number of Onto functions | Download |

198 | Number of Bijections | Download |

199 | Counting number of functions. | Download |

200 | Motivation for Composition of functions - Part 1 | Download |

201 | Motivation for Composition of functions - Part 2 | Download |

202 | Definition of Composition of functions | Download |

203 | Why study Composition of functions | Download |

204 | Example of Composition of functions - Part 1 | Download |

205 | Example of Composition of functions - Part 2 | Download |

206 | Motivation for Inverse functions | Download |

207 | Inverse functions | Download |

208 | Examples of Inverse functions | Download |

209 | Application of inverse functions - Part 1 | Download |

210 | Three stories | Download |

211 | Three stories - Connecting the dots | Download |

212 | Mathematical induction - An illustration | Download |

213 | Mathematical Induction - Its essence | Download |

214 | Mathematical Induction - The formal way | Download |

215 | MI - Sum of odd numbers | Download |

216 | MI - Sum of powers of 2 | Download |

217 | MI - Inequality 1 | Download |

218 | MI - Inequality 1 (solution) | Download |

219 | MI - To prove divisibility | Download |

220 | MI - To prove divisibility (solution) | Download |

221 | MI - Problem on satisfying inequalities | Download |

222 | MI - Problem on satisfying inequalities (solutions) | Download |

223 | MI - Inequality 2 | Download |

224 | MI - Inequality 2 solution | Download |

225 | Mathematical Induction - Example 9 | Download |

226 | Mathematical Induction - Example 10 solution | Download |

227 | Binomial Coeffecients - Proof by induction | Download |

228 | Checker board and Triomioes - A puzzle | Download |

229 | Checker board and triominoes - Solution | Download |

230 | Mathematical induction - An important note | Download |

231 | Mathematical Induction - A false proof | Download |

232 | A false proof - Solution | Download |

233 | Motivation for Pegionhole Principle | Download |

234 | Group of n people | Download |

235 | Set of n integgers | Download |

236 | 10 points on an equilateral triangle | Download |

237 | Pegionhole Principle - A result | Download |

238 | Consecutive integers | Download |

239 | Consecutive integers solution | Download |

240 | Matching initials | Download |

241 | Matching initials - Solution | Download |

242 | Numbers adding to 9 | Download |

243 | Numbers adding to 9 - Solution | Download |

244 | Deck of cards | Download |

245 | Deck of cards - Solution | Download |

246 | Number of errors | Download |

247 | Number of errors - Solution | Download |

248 | Puzzle - Challenge for you | Download |

249 | Friendship - an interesting property | Download |

250 | Connectedness through Connecting people | Download |

251 | Traversing the bridges | Download |

252 | Three utilities problem | Download |

253 | Coloring the India map | Download |

254 | Defintion of a Graph | Download |

255 | Degree and degree sequence | Download |

256 | Relation between number of edges and degrees | Download |

257 | Relation between number of edges and degrees - Proof | Download |

258 | Hand shaking lemma - Corollary | Download |

259 | Problems based on Hand shaking lemma | Download |

260 | Havel Hakimi theorem - Part 1 | Download |

261 | Havel Hakimi theorem - Part 2 | Download |

262 | Havel Hakimi theorem - Part 3 | Download |

263 | Havel Hakimi theorem - Part 4 | Download |

264 | Havel Hakimi theorem - Part 5 | Download |

265 | Regular graph and irregular graph | Download |

266 | Walk | Download |

267 | Trail | Download |

268 | Path and closed path | Download |

269 | Definitions revisited | Download |

270 | Examples of walk, trail and path | Download |

271 | Cycle and circuit | Download |

272 | Example of cycle and circuit | Download |

273 | Relation between walk and path | Download |

274 | Relation between walk and path - An induction proof | Download |

275 | Subgraph | Download |

276 | Spanning and induced subgraph | Download |

277 | Spanning and induced subgraph - A result | Download |

278 | Introduction to Tree | Download |

279 | Connected and Disconnected graphs | Download |

280 | Property of a cycle | Download |

281 | Edge condition for connectivity | Download |

282 | Connecting connectedness and path | Download |

283 | Connecting connectedness and path - An illustration | Download |

284 | Cut vertex | Download |

285 | Cut edge | Download |

286 | Illustration of cut vertices and cut edges | Download |

287 | NetworkX - Need of the hour | Download |

288 | Introduction to Python - Installation | Download |

289 | Introduction to Python - Basics | Download |

290 | Introduction to NetworkX | Download |

291 | Story so far - Using NetworkX | Download |

292 | Directed, weighted and multi graphs | Download |

293 | Illustration of Directed, weighted and multi graphs | Download |

294 | Graph representations - Introduction | Download |

295 | Adjacency matrix representation | Download |

296 | Incidence matrix representation | Download |

297 | Isomorphism - Introduction | Download |

298 | Isomorphic graphs - An illustration | Download |

299 | Isomorphic graphs - A challenge | Download |

300 | Non - isomorphic graphs | Download |

301 | Isomorphism - A question | Download |

302 | Complement of a Graph - Introduction | Download |

303 | Complement of a Graph - Illiustration | Download |

304 | Self complement | Download |

305 | Complement of a disconnected graph is connected | Download |

306 | Complement of a disconnected graph is connected - Solution | Download |

307 | Which is more? Connected graphs or disconnected graphs? | Download |

308 | Bipartite graphs | Download |

309 | Bipartite graphs - A puzzle | Download |

310 | Bipartite graphs - Converse part of the puzzle | Download |

311 | Definition of Eulerian Graph | Download |

312 | Illustration of eulerian graph | Download |

313 | Non- example of Eulerian graph | Download |

314 | Litmus test for an Eulerian graph | Download |

315 | Why even degree? | Download |

316 | Proof for even degree implies graph is eulerian | Download |

317 | A condition for Eulerian trail | Download |

318 | Why the name Eulerian | Download |

319 | Can you traverse all location? | Download |

320 | Defintion of Hamiltonian graphs | Download |

321 | Examples of Hamiltonian graphs | Download |

322 | Hamiltonian graph - A result | Download |

323 | A result on connectedness | Download |

324 | A result on Path | Download |

325 | Dirac's Theorem | Download |

326 | Dirac's theorem - A note | Download |

327 | Ore's Theorem | Download |

328 | Dirac's Theorem v/s Ore's Theorem | Download |

329 | Eulerian and Hamiltonian Are they related | Download |

330 | Importance of Hamiltonian graphs in Computer science | Download |

331 | Constructing non intersecting roads | Download |

332 | Definition of a Planar graph | Download |

333 | Examples of Planar graphs | Download |

334 | V - E + R = 2 | Download |

335 | Illustration of V - E + R =2 | Download |

336 | V - E + R = 2; Use induction | Download |

337 | Proof of V - E + R = 2 | Download |

338 | Famous non-planar graphs | Download |

339 | Litmus test for planarity | Download |

340 | Planar graphs - Inequality 1 | Download |

341 | 3 Utilities problem - Revisited | Download |

342 | Complete graph on 5 vertices is non-planar - Proof | Download |

343 | Prisoners and cells | Download |

344 | Prisoners example and Proper coloring | Download |

345 | Chromatic number of a graph | Download |

346 | Examples on Proper coloring | Download |

347 | Recalling the India map problem | Download |

348 | Recalling the India map problem - Solution | Download |

349 | NetworkX - Digraphs | Download |

350 | NetworkX - Adjacency matrix | Download |

351 | NetworkX- Random graphs | Download |

352 | NetworkX - Subgarph | Download |

353 | NetworkX - Isomorphic graphs Part 1 | Download |

354 | NetworkX - Isomorphic graphs Part 2 | Download |

355 | NetworkX - Isomorphic graphs: A game to play | Download |

356 | NetworkX - Graph complement | Download |

357 | NetworkX - Eulerian graphs | Download |

358 | NetworkX - Bipaprtite graphs | Download |

359 | NetworkX - Coloring | Download |

360 | Counting in a creative way | Download |

361 | Example 1 - Fun with words | Download |

362 | Words and the polynomial | Download |

363 | Words and the polynomial - Explained | Download |

364 | Example 2 - Picking five balls | Download |

365 | Picking five balls - Solution | Download |

366 | Picking five balls - Another version | Download |

367 | Defintion of Generating function | Download |

368 | Generating function examples - Part 1 | Download |

369 | Generating function examples - Part 2 | Download |

370 | Generating function examples - Part 3 | Download |

371 | Binomial expansion - A generating function | Download |

372 | Binomial expansion - Explained | Download |

373 | Picking 7 balls - The naive way | Download |

374 | Picking 7 balls - The creative way | Download |

375 | Generating functions - Problem 1 | Download |

376 | Generating functions - Problem 2 | Download |

377 | Generating functions - Problem 3 | Download |

378 | Why Generating function? | Download |

379 | Introduction to Advanced Counting | Download |

380 | Example 1 : Dogs and Cats | Download |

381 | Inclusion-Exclusion Formula | Download |

382 | Proof of Inclusion - Exlusion formula | Download |

383 | Example 2 : Integer solutions of an equation | Download |

384 | Example 3 : Words not containing some strings | Download |

385 | Example 4 : Arranging 3 x's, 3 y's and 3 z's | Download |

386 | Example 5 : Non-multiples of 2 or 3 | Download |

387 | Example 6 : Integers not divisible by 5, 7 or 11 | Download |

388 | A tip in solving problems | Download |

389 | Example 7 : A dog nor a cat | Download |

390 | Example 8 : Brownies, Muffins and Cookies | Download |

391 | Example 10 : Integer solutions of an equation | Download |

392 | Example 11 : Seating Arrangement - Part 1 | Download |

393 | Example 11 : Seating Arrangement - Part 2 | Download |

394 | Example 12 : Integer solutions of an equation | Download |

395 | Number of Onto Functions. | Download |

396 | Formula for Number of Onto Functions | Download |

397 | Example 13 : Onto Functions | Download |

398 | Example 14 : No one in their own house | Download |

399 | Derangements | Download |

400 | Derangements of 4 numbers | Download |

401 | Example 15 : Bottles and caps | Download |

402 | Example 16 : Self grading | Download |

403 | Example 17 : Even integers and their places | Download |

404 | Example 18 : Finding total number of items | Download |

405 | Example 19 : Devising a secret code | Download |

406 | Placing rooks on the chessboard | Download |

407 | Rook Polynomial | Download |

408 | Rook Polynomial. | Download |

409 | Motivation for recurrence relation | Download |

410 | Getting started with recurrence relations | Download |

411 | What is a recurrence relation? | Download |

412 | Compound Interest as a recurrence relation | Download |

413 | Examples of recurrence relations | Download |

414 | Example - Number of ways of climbing steps | Download |

415 | Number of ways of climbing steps: Recurrence relation | Download |

416 | Example - Rabbits on an island | Download |

417 | Example - n-bit string | Download |

418 | Example - n-bit string without consecutive zero | Download |

419 | Solving Linear Recurrence Relations - A theorem | Download |

420 | A note on the proof | Download |

421 | Soving recurrence relation - Example 1 | Download |

422 | Soving recurrence relation - Example 2 | Download |

423 | Fibonacci Sequence | Download |

424 | Introduction to Fibonacci sequence | Download |

425 | Solution of Fibbonacci sequence | Download |

426 | A basic introduction to 'complexity' | Download |

427 | Intuition for 'complexity' | Download |

428 | Visualizing complexity order as a graph | Download |

429 | Tower of Hanoi | Download |

430 | Reccurence relation of Tower of Hanoi | Download |

431 | Solution for the recurrence relation of Tower of Hanoi | Download |

432 | A searching technique | Download |

433 | Recurrence relation for Binary search | Download |

434 | Solution for the recurrence relation of Binary search | Download |

435 | Example: Door knocks example | Download |

436 | Example: Door knocks example solution | Download |

437 | Door knock example and Merge sort | Download |

438 | Introduction to Merge sort - 1 | Download |

439 | Recurrence relation for Merge sort | Download |

440 | Intoduction to advanced topics | Download |

441 | Introduction to Chromatic polynomial | Download |

442 | Chromatic polynomial of complete graphs | Download |

443 | Chromatic polynomial of cycle on 4 vertices - Part 1 | Download |

444 | Chromatic polynomial of cycle on 4 vertices - Part 2 | Download |

445 | Correspondence between partition and generating functions | Download |

446 | Correspondence between partition and generating functions: In general | Download |

447 | Distinct partitions and odd partitions | Download |

448 | Distinct partitions and generating functions | Download |

449 | Odd partitions and generating functions | Download |

450 | Distinct partitions equals odd partitions: Observation | Download |

451 | Distinct partitions equals odd partitions: Proof | Download |

452 | Why 'partitions' to 'polynomial'? | Download |

453 | Example: Picking 4 letters from the word 'INDIAN' | Download |

454 | Motivation for exponential generating function | Download |

455 | Recurrrence relation: The theorem and its proof | Download |

456 | Introduction to Group Theory | Download |

457 | Uniqueness of the identity element | Download |

458 | Formal definition of a Group | Download |

459 | Groups: Examples and non-examples | Download |

460 | Groups: Special Examples Part 1 | Download |

461 | Groups: Special Examples Part 2 | Download |

462 | Subgroup: Defintion and examples | Download |

463 | Lagrange's theorem | Download |

464 | Summary. | Download |

465 | Conclusion. | Download |

Sl.No | Chapter Name | English |
---|---|---|

1 | Motivation for Counting | Download Verified |

2 | Paper Folding Example | Download Verified |

3 | Rubik's Cube Example | Download Verified |

4 | Factorial Example | Download Verified |

5 | Counting in Computer Science | Download Verified |

6 | Motivation for Catalan numbers | Download Verified |

7 | Rule of Sum and Rule of Product | Download Verified |

8 | Problems on Rule of Sum and Rule of Product | Download Verified |

9 | Factorial Explained | Download Verified |

10 | Proof of n! - Part 1 | Download Verified |

11 | Proof of n! - Part 2 | Download Verified |

12 | Astronomical Numbers | Download Verified |

13 | Permutations - Part 1 | Download Verified |

14 | Permutations - Part 2 | Download Verified |

15 | Permutations - Part 3 | Download Verified |

16 | Permutations - Part 4 | Download Verified |

17 | Problems on Permutations | Download Verified |

18 | Combinations - Part 1 | Download Verified |

19 | Combinations - Part 2 | Download Verified |

20 | Combinations - Part 3 | Download Verified |

21 | Combinations - Part 4 | Download Verified |

22 | Problems on Combinations | Download Verified |

23 | Difference between Permuations and Combinations | Download Verified |

24 | Combination with Repetition - Part 1 | Download Verified |

25 | Combination with Repetition - Part 2 | Download Verified |

26 | Combination with Repetition - Problems | Download Verified |

27 | Binomial theorem | Download Verified |

28 | Applications of Binomial theorem | Download Verified |

29 | Properties of Binomial theorem | Download Verified |

30 | Multinomial theorem | Download Verified |

31 | Problems on Binomial theorem | Download Verified |

32 | Pascal's Triangle | Download Verified |

33 | Fun facts on Pascal's Triangle | Download Verified |

34 | Catalan Numbers - Part 1 | Download Verified |

35 | Catalan Numbers - Part 2 | Download Verified |

36 | Catalan Numbers - Part 3 | Download Verified |

37 | Catalan Numbers - Part 4 | Download Verified |

38 | Examples of Catalan numbers | Download Verified |

39 | Chapter Summary | Download Verified |

40 | Introduction to Set Theory | Download Verified |

41 | Example, definiton and notation | Download Verified |

42 | Sets - Problems Part 1 | Download Verified |

43 | Subsets - Part 1 | Download Verified |

44 | Subsets - Part 2 | Download Verified |

45 | Subsets - Part 3 | Download Verified |

46 | Union and intersections of sets | Download Verified |

47 | Union and intersections of sets - Part 1 | Download Verified |

48 | Union and intersections of sets - Part 2 | Download Verified |

49 | Union and intersections of sets - Part 3 | Download Verified |

50 | Cardinality of Union of two sets - Part 1 | Download Verified |

51 | Cardinality of Union of sets - Part 2 | Download Verified |

52 | Cardinality of Union of three sets | Download Verified |

53 | Power Set - Part 1 | Download Verified |

54 | Power set - Part 2 | Download Verified |

55 | Power set - Part 3 | Download Verified |

56 | Connection betwenn Binomial Theorem and Power Sets | Download Verified |

57 | Power set - Problems | Download Verified |

58 | Complement of a set | Download Verified |

59 | De Morgan's Laws - Part 1 | Download Verified |

60 | De Morgan's Laws - Part 2 | Download Verified |

61 | A proof technique | Download Verified |

62 | De Morgan's Laws - Part 3 | Download Verified |

63 | De Morgan's Laws - Part 4 | Download Verified |

64 | Set difference - Part 1 | Download Verified |

65 | Set difference - Part 2 | Download Verified |

66 | Symmetric difference | Download Verified |

67 | History | Download Verified |

68 | Summary | Download Verified |

69 | Motivational example | Download Verified |

70 | Introduction to Statements | Download Verified |

71 | Examples and Non-examples of Statements | Download Verified |

72 | Introduction to Negation | Download Verified |

73 | Negation - Explanation | Download Verified |

74 | Negation - Truthtable | Download Verified |

75 | Examples for Negation | Download Verified |

76 | Motivation for OR operator | Download Verified |

77 | Introduction to OR operator | Download Verified |

78 | Truthtable for OR operator | Download Verified |

79 | OR operator for 3 Variables | Download Verified |

80 | Truthtable for AND operator | Download Verified |

81 | AND operator for 3 Variables | Download Verified |

82 | Primitive and Compound statements - Part 1 | Download Verified |

83 | Primitive and Compound statements - Part 2 | Download Verified |

84 | Problems involoving NOT, OR and AND operators | Download Verified |

85 | Introduction to implication | Download Verified |

86 | Examples and Non-examples of Implication - Part 1 | Download Verified |

87 | Examples and Non-examples of Implication - Part 2 | Download Verified |

88 | Explanation of Implication | Download Verified |

89 | Introduction to Double Implication | Download Verified |

90 | Explanation of Double Implication | Download Verified |

91 | Converse, Inverse and Contrapositive | Download Verified |

92 | XOR operator - Part 1 | Download Verified |

93 | XOR operator - Part 2 | Download Verified |

94 | XOR operator - Part 3 | Download Verified |

95 | Problems | Download Verified |

96 | Tautology, Contradiction - Part 1 | Download Verified |

97 | Tautology, Contradiction - Part 2 | Download Verified |

98 | Tautology, Contradiction - Part 3 | Download Verified |

99 | SAT Problem - Part 1 | Download Verified |

100 | SAT Problem - Part 2 | Download Verified |

101 | Logical Equivalence - Part 1 | Download Verified |

102 | Logical Equivalence - Part 2 | Download Verified |

103 | Logical Equivalence - Part 3 | Download Verified |

104 | Logical Equivalence - Part 4 | Download Verified |

105 | Motivation for laws of logic | Download Verified |

106 | Double negation - Part 1 | Download Verified |

107 | Double negation - Part 2 | Download Verified |

108 | Laws of Logic | Download Verified |

109 | De Morgan's Law - Part 1 | Download Verified |

110 | De Morgan's Law - Part 2 | Download Verified |

111 | Rules of Inferences - Part 1 | Download Verified |

112 | Rules of Inferences - Part 2 | Download Verified |

113 | Rules of Inferences - Part 3 | Download Verified |

114 | Rules of Inferences - Part 4 | Download Verified |

115 | Rules of Inferences - Part 5 | Download Verified |

116 | Rules of Inferences - Part 6 | Download Verified |

117 | Rules of Inferences - Part 7 | Download Verified |

118 | Conclusion | Download Verified |

119 | Introduction to Relation | Download Verified |

120 | Graphical Representation of a Relation | Download Verified |

121 | Various sets | Download Verified |

122 | Matrix Representation of a Relation | Download Verified |

123 | Relation - An Example | Download Verified |

124 | Cartesian Product | Download Verified |

125 | Set Representation of a Relation | Download Verified |

126 | Revisiting Representations of a Relation | Download Verified |

127 | Examples of Relations | Download Verified |

128 | Number of relations - Part 1 | Download Verified |

129 | Number of relations - Part 2 | Download Verified |

130 | Reflexive relation - Introduction | Download Verified |

131 | Example of a Reflexive relation | Download Verified |

132 | Reflexive relation - Matrix representation | Download Verified |

133 | Number of Reflexive relations | Download Verified |

134 | Symmetric Relation - Introduction | Download Verified |

135 | Symmetric Relation - Matrix representation | Download Verified |

136 | Symmetric Relation - Examples and non examples | Download Verified |

137 | Parallel lines revisited | Download Verified |

138 | Number of symmetric relations - Part 1 | Download Verified |

139 | Number of symmetric relations - Part 2 | Download Verified |

140 | Examples of Reflexive and Symmetric Relations | Download Verified |

141 | Pattern | Download Verified |

142 | Transitive relation - Examples and non examples | Download Verified |

143 | Antisymmetric relation | Download Verified |

144 | Examples of Transitive and Antisymmetric Relation | Download Verified |

145 | Antisymmetric - Graphical representation | Download Verified |

146 | Antisymmetric - Matrix representation | Download Verified |

147 | Number of Antisymmetric relations | Download Verified |

148 | Condition for relation to be reflexive | Download Verified |

149 | Few notations | Download Verified |

150 | Condition for relation to be reflexive. | Download Verified |

151 | Condition for relation to be reflexive.. | Download Verified |

152 | Condition for relation to be symmetric | Download Verified |

153 | Condition for relation to be symmetric. | Download Verified |

154 | Condition for relation to be antisymmetric | Download Verified |

155 | Equivalence relation | Download Verified |

156 | Equivalence relation - Example 4 | Download Verified |

157 | Partition - Part 1 | Download Verified |

158 | Partition - Part 2 | Download Verified |

159 | Partition - Part 3 | Download Verified |

160 | Partition - Part 4 | Download Verified |

161 | Partition - Part 5 | Download Verified |

162 | Partition - Part 5. | PDF unavailable |

163 | Motivational Example - 1 | PDF unavailable |

164 | Motivational Example - 2 | PDF unavailable |

165 | Commonality in examples | PDF unavailable |

166 | Motivational Example - 3 | PDF unavailable |

167 | Example - 4 Explanation | PDF unavailable |

168 | Introduction to functions | PDF unavailable |

169 | Defintion of a function - Part 1 | PDF unavailable |

170 | Defintion of a function - Part 2 | PDF unavailable |

171 | Defintion of a function - Part 3 | PDF unavailable |

172 | Relations vs Functions - Part 1 | PDF unavailable |

173 | Relations vs Functions - Part 2 | PDF unavailable |

174 | Introduction to One-One Function | PDF unavailable |

175 | One-One Function - Example 1 | PDF unavailable |

176 | One-One Function - Example 2 | PDF unavailable |

177 | One-One Function - Example 3 | PDF unavailable |

178 | Proving a Function is One-One | PDF unavailable |

179 | Examples and Non- examples of One-One function | PDF unavailable |

180 | Cardinality condition in One-One function - Part 1 | PDF unavailable |

181 | Cardinality condition in One-One function - Part 2 | PDF unavailable |

182 | Introduction to Onto Function - Part 1 | PDF unavailable |

183 | Introduction to Onto Function - Part 2 | PDF unavailable |

184 | Definition of Onto Function | PDF unavailable |

185 | Examples of Onto Function | PDF unavailable |

186 | Cardinality condition in Onto function - Part 1 | PDF unavailable |

187 | Cardinality condition in Onto function - Part 2 | PDF unavailable |

188 | Introduction to Bijection | PDF unavailable |

189 | Examples of Bijection | PDF unavailable |

190 | Cardinality condition in Bijection - Part 1 | PDF unavailable |

191 | Cardinality condition in Bijection - Part 2 | PDF unavailable |

192 | Counting number of functions | PDF unavailable |

193 | Number of functions | PDF unavailable |

194 | Number of One-One functions - Part 1 | PDF unavailable |

195 | Number of One-One functions - Part 2 | PDF unavailable |

196 | Number of One-One functions - Part 3 | PDF unavailable |

197 | Number of Onto functions | PDF unavailable |

198 | Number of Bijections | PDF unavailable |

199 | Counting number of functions. | PDF unavailable |

200 | Motivation for Composition of functions - Part 1 | PDF unavailable |

201 | Motivation for Composition of functions - Part 2 | PDF unavailable |

202 | Definition of Composition of functions | PDF unavailable |

203 | Why study Composition of functions | PDF unavailable |

204 | Example of Composition of functions - Part 1 | PDF unavailable |

205 | Example of Composition of functions - Part 2 | PDF unavailable |

206 | Motivation for Inverse functions | PDF unavailable |

207 | Inverse functions | PDF unavailable |

208 | Examples of Inverse functions | PDF unavailable |

209 | Application of inverse functions - Part 1 | PDF unavailable |

210 | Three stories | PDF unavailable |

211 | Three stories - Connecting the dots | PDF unavailable |

212 | Mathematical induction - An illustration | PDF unavailable |

213 | Mathematical Induction - Its essence | PDF unavailable |

214 | Mathematical Induction - The formal way | PDF unavailable |

215 | MI - Sum of odd numbers | PDF unavailable |

216 | MI - Sum of powers of 2 | PDF unavailable |

217 | MI - Inequality 1 | PDF unavailable |

218 | MI - Inequality 1 (solution) | PDF unavailable |

219 | MI - To prove divisibility | PDF unavailable |

220 | MI - To prove divisibility (solution) | PDF unavailable |

221 | MI - Problem on satisfying inequalities | PDF unavailable |

222 | MI - Problem on satisfying inequalities (solutions) | PDF unavailable |

223 | MI - Inequality 2 | PDF unavailable |

224 | MI - Inequality 2 solution | PDF unavailable |

225 | Mathematical Induction - Example 9 | PDF unavailable |

226 | Mathematical Induction - Example 10 solution | PDF unavailable |

227 | Binomial Coeffecients - Proof by induction | PDF unavailable |

228 | Checker board and Triomioes - A puzzle | PDF unavailable |

229 | Checker board and triominoes - Solution | PDF unavailable |

230 | Mathematical induction - An important note | PDF unavailable |

231 | Mathematical Induction - A false proof | PDF unavailable |

232 | A false proof - Solution | PDF unavailable |

233 | Motivation for Pegionhole Principle | PDF unavailable |

234 | Group of n people | PDF unavailable |

235 | Set of n integgers | PDF unavailable |

236 | 10 points on an equilateral triangle | PDF unavailable |

237 | Pegionhole Principle - A result | PDF unavailable |

238 | Consecutive integers | PDF unavailable |

239 | Consecutive integers solution | PDF unavailable |

240 | Matching initials | PDF unavailable |

241 | Matching initials - Solution | PDF unavailable |

242 | Numbers adding to 9 | PDF unavailable |

243 | Numbers adding to 9 - Solution | PDF unavailable |

244 | Deck of cards | PDF unavailable |

245 | Deck of cards - Solution | PDF unavailable |

246 | Number of errors | PDF unavailable |

247 | Number of errors - Solution | PDF unavailable |

248 | Puzzle - Challenge for you | PDF unavailable |

249 | Friendship - an interesting property | PDF unavailable |

250 | Connectedness through Connecting people | PDF unavailable |

251 | Traversing the bridges | PDF unavailable |

252 | Three utilities problem | PDF unavailable |

253 | Coloring the India map | PDF unavailable |

254 | Defintion of a Graph | PDF unavailable |

255 | Degree and degree sequence | PDF unavailable |

256 | Relation between number of edges and degrees | PDF unavailable |

257 | Relation between number of edges and degrees - Proof | PDF unavailable |

258 | Hand shaking lemma - Corollary | PDF unavailable |

259 | Problems based on Hand shaking lemma | PDF unavailable |

260 | Havel Hakimi theorem - Part 1 | PDF unavailable |

261 | Havel Hakimi theorem - Part 2 | PDF unavailable |

262 | Havel Hakimi theorem - Part 3 | PDF unavailable |

263 | Havel Hakimi theorem - Part 4 | PDF unavailable |

264 | Havel Hakimi theorem - Part 5 | PDF unavailable |

265 | Regular graph and irregular graph | PDF unavailable |

266 | Walk | PDF unavailable |

267 | Trail | PDF unavailable |

268 | Path and closed path | PDF unavailable |

269 | Definitions revisited | PDF unavailable |

270 | Examples of walk, trail and path | PDF unavailable |

271 | Cycle and circuit | PDF unavailable |

272 | Example of cycle and circuit | PDF unavailable |

273 | Relation between walk and path | PDF unavailable |

274 | Relation between walk and path - An induction proof | PDF unavailable |

275 | Subgraph | PDF unavailable |

276 | Spanning and induced subgraph | PDF unavailable |

277 | Spanning and induced subgraph - A result | PDF unavailable |

278 | Introduction to Tree | PDF unavailable |

279 | Connected and Disconnected graphs | PDF unavailable |

280 | Property of a cycle | PDF unavailable |

281 | Edge condition for connectivity | PDF unavailable |

282 | Connecting connectedness and path | PDF unavailable |

283 | Connecting connectedness and path - An illustration | PDF unavailable |

284 | Cut vertex | PDF unavailable |

285 | Cut edge | PDF unavailable |

286 | Illustration of cut vertices and cut edges | PDF unavailable |

287 | NetworkX - Need of the hour | PDF unavailable |

288 | Introduction to Python - Installation | PDF unavailable |

289 | Introduction to Python - Basics | PDF unavailable |

290 | Introduction to NetworkX | PDF unavailable |

291 | Story so far - Using NetworkX | PDF unavailable |

292 | Directed, weighted and multi graphs | PDF unavailable |

293 | Illustration of Directed, weighted and multi graphs | PDF unavailable |

294 | Graph representations - Introduction | PDF unavailable |

295 | Adjacency matrix representation | PDF unavailable |

296 | Incidence matrix representation | PDF unavailable |

297 | Isomorphism - Introduction | PDF unavailable |

298 | Isomorphic graphs - An illustration | PDF unavailable |

299 | Isomorphic graphs - A challenge | PDF unavailable |

300 | Non - isomorphic graphs | PDF unavailable |

301 | Isomorphism - A question | PDF unavailable |

302 | Complement of a Graph - Introduction | PDF unavailable |

303 | Complement of a Graph - Illiustration | PDF unavailable |

304 | Self complement | PDF unavailable |

305 | Complement of a disconnected graph is connected | PDF unavailable |

306 | Complement of a disconnected graph is connected - Solution | PDF unavailable |

307 | Which is more? Connected graphs or disconnected graphs? | PDF unavailable |

308 | Bipartite graphs | PDF unavailable |

309 | Bipartite graphs - A puzzle | PDF unavailable |

310 | Bipartite graphs - Converse part of the puzzle | PDF unavailable |

311 | Definition of Eulerian Graph | PDF unavailable |

312 | Illustration of eulerian graph | PDF unavailable |

313 | Non- example of Eulerian graph | PDF unavailable |

314 | Litmus test for an Eulerian graph | PDF unavailable |

315 | Why even degree? | PDF unavailable |

316 | Proof for even degree implies graph is eulerian | PDF unavailable |

317 | A condition for Eulerian trail | PDF unavailable |

318 | Why the name Eulerian | PDF unavailable |

319 | Can you traverse all location? | PDF unavailable |

320 | Defintion of Hamiltonian graphs | PDF unavailable |

321 | Examples of Hamiltonian graphs | PDF unavailable |

322 | Hamiltonian graph - A result | PDF unavailable |

323 | A result on connectedness | PDF unavailable |

324 | A result on Path | PDF unavailable |

325 | Dirac's Theorem | PDF unavailable |

326 | Dirac's theorem - A note | PDF unavailable |

327 | Ore's Theorem | PDF unavailable |

328 | Dirac's Theorem v/s Ore's Theorem | PDF unavailable |

329 | Eulerian and Hamiltonian Are they related | PDF unavailable |

330 | Importance of Hamiltonian graphs in Computer science | PDF unavailable |

331 | Constructing non intersecting roads | PDF unavailable |

332 | Definition of a Planar graph | PDF unavailable |

333 | Examples of Planar graphs | PDF unavailable |

334 | V - E + R = 2 | PDF unavailable |

335 | Illustration of V - E + R =2 | PDF unavailable |

336 | V - E + R = 2; Use induction | PDF unavailable |

337 | Proof of V - E + R = 2 | PDF unavailable |

338 | Famous non-planar graphs | PDF unavailable |

339 | Litmus test for planarity | PDF unavailable |

340 | Planar graphs - Inequality 1 | PDF unavailable |

341 | 3 Utilities problem - Revisited | PDF unavailable |

342 | Complete graph on 5 vertices is non-planar - Proof | PDF unavailable |

343 | Prisoners and cells | PDF unavailable |

344 | Prisoners example and Proper coloring | PDF unavailable |

345 | Chromatic number of a graph | PDF unavailable |

346 | Examples on Proper coloring | PDF unavailable |

347 | Recalling the India map problem | PDF unavailable |

348 | Recalling the India map problem - Solution | PDF unavailable |

349 | NetworkX - Digraphs | PDF unavailable |

350 | NetworkX - Adjacency matrix | PDF unavailable |

351 | NetworkX- Random graphs | PDF unavailable |

352 | NetworkX - Subgarph | PDF unavailable |

353 | NetworkX - Isomorphic graphs Part 1 | PDF unavailable |

354 | NetworkX - Isomorphic graphs Part 2 | PDF unavailable |

355 | NetworkX - Isomorphic graphs: A game to play | PDF unavailable |

356 | NetworkX - Graph complement | PDF unavailable |

357 | NetworkX - Eulerian graphs | PDF unavailable |

358 | NetworkX - Bipaprtite graphs | PDF unavailable |

359 | NetworkX - Coloring | PDF unavailable |

360 | Counting in a creative way | PDF unavailable |

361 | Example 1 - Fun with words | PDF unavailable |

362 | Words and the polynomial | PDF unavailable |

363 | Words and the polynomial - Explained | PDF unavailable |

364 | Example 2 - Picking five balls | PDF unavailable |

365 | Picking five balls - Solution | PDF unavailable |

366 | Picking five balls - Another version | PDF unavailable |

367 | Defintion of Generating function | PDF unavailable |

368 | Generating function examples - Part 1 | PDF unavailable |

369 | Generating function examples - Part 2 | PDF unavailable |

370 | Generating function examples - Part 3 | PDF unavailable |

371 | Binomial expansion - A generating function | PDF unavailable |

372 | Binomial expansion - Explained | PDF unavailable |

373 | Picking 7 balls - The naive way | PDF unavailable |

374 | Picking 7 balls - The creative way | PDF unavailable |

375 | Generating functions - Problem 1 | PDF unavailable |

376 | Generating functions - Problem 2 | PDF unavailable |

377 | Generating functions - Problem 3 | PDF unavailable |

378 | Why Generating function? | PDF unavailable |

379 | Introduction to Advanced Counting | PDF unavailable |

380 | Example 1 : Dogs and Cats | PDF unavailable |

381 | Inclusion-Exclusion Formula | PDF unavailable |

382 | Proof of Inclusion - Exlusion formula | PDF unavailable |

383 | Example 2 : Integer solutions of an equation | PDF unavailable |

384 | Example 3 : Words not containing some strings | PDF unavailable |

385 | Example 4 : Arranging 3 x's, 3 y's and 3 z's | PDF unavailable |

386 | Example 5 : Non-multiples of 2 or 3 | PDF unavailable |

387 | Example 6 : Integers not divisible by 5, 7 or 11 | PDF unavailable |

388 | A tip in solving problems | PDF unavailable |

389 | Example 7 : A dog nor a cat | PDF unavailable |

390 | Example 8 : Brownies, Muffins and Cookies | PDF unavailable |

391 | Example 10 : Integer solutions of an equation | PDF unavailable |

392 | Example 11 : Seating Arrangement - Part 1 | PDF unavailable |

393 | Example 11 : Seating Arrangement - Part 2 | PDF unavailable |

394 | Example 12 : Integer solutions of an equation | PDF unavailable |

395 | Number of Onto Functions. | PDF unavailable |

396 | Formula for Number of Onto Functions | PDF unavailable |

397 | Example 13 : Onto Functions | PDF unavailable |

398 | Example 14 : No one in their own house | PDF unavailable |

399 | Derangements | PDF unavailable |

400 | Derangements of 4 numbers | PDF unavailable |

401 | Example 15 : Bottles and caps | PDF unavailable |

402 | Example 16 : Self grading | PDF unavailable |

403 | Example 17 : Even integers and their places | PDF unavailable |

404 | Example 18 : Finding total number of items | PDF unavailable |

405 | Example 19 : Devising a secret code | PDF unavailable |

406 | Placing rooks on the chessboard | PDF unavailable |

407 | Rook Polynomial | PDF unavailable |

408 | Rook Polynomial. | PDF unavailable |

409 | Motivation for recurrence relation | PDF unavailable |

410 | Getting started with recurrence relations | PDF unavailable |

411 | What is a recurrence relation? | PDF unavailable |

412 | Compound Interest as a recurrence relation | PDF unavailable |

413 | Examples of recurrence relations | PDF unavailable |

414 | Example - Number of ways of climbing steps | PDF unavailable |

415 | Number of ways of climbing steps: Recurrence relation | PDF unavailable |

416 | Example - Rabbits on an island | PDF unavailable |

417 | Example - n-bit string | PDF unavailable |

418 | Example - n-bit string without consecutive zero | PDF unavailable |

419 | Solving Linear Recurrence Relations - A theorem | PDF unavailable |

420 | A note on the proof | PDF unavailable |

421 | Soving recurrence relation - Example 1 | PDF unavailable |

422 | Soving recurrence relation - Example 2 | PDF unavailable |

423 | Fibonacci Sequence | PDF unavailable |

424 | Introduction to Fibonacci sequence | PDF unavailable |

425 | Solution of Fibbonacci sequence | PDF unavailable |

426 | A basic introduction to 'complexity' | PDF unavailable |

427 | Intuition for 'complexity' | PDF unavailable |

428 | Visualizing complexity order as a graph | PDF unavailable |

429 | Tower of Hanoi | PDF unavailable |

430 | Reccurence relation of Tower of Hanoi | PDF unavailable |

431 | Solution for the recurrence relation of Tower of Hanoi | PDF unavailable |

432 | A searching technique | PDF unavailable |

433 | Recurrence relation for Binary search | PDF unavailable |

434 | Solution for the recurrence relation of Binary search | PDF unavailable |

435 | Example: Door knocks example | PDF unavailable |

436 | Example: Door knocks example solution | PDF unavailable |

437 | Door knock example and Merge sort | PDF unavailable |

438 | Introduction to Merge sort - 1 | PDF unavailable |

439 | Recurrence relation for Merge sort | PDF unavailable |

440 | Intoduction to advanced topics | PDF unavailable |

441 | Introduction to Chromatic polynomial | PDF unavailable |

442 | Chromatic polynomial of complete graphs | PDF unavailable |

443 | Chromatic polynomial of cycle on 4 vertices - Part 1 | PDF unavailable |

444 | Chromatic polynomial of cycle on 4 vertices - Part 2 | PDF unavailable |

445 | Correspondence between partition and generating functions | PDF unavailable |

446 | Correspondence between partition and generating functions: In general | PDF unavailable |

447 | Distinct partitions and odd partitions | PDF unavailable |

448 | Distinct partitions and generating functions | PDF unavailable |

449 | Odd partitions and generating functions | PDF unavailable |

450 | Distinct partitions equals odd partitions: Observation | PDF unavailable |

451 | Distinct partitions equals odd partitions: Proof | PDF unavailable |

452 | Why 'partitions' to 'polynomial'? | PDF unavailable |

453 | Example: Picking 4 letters from the word 'INDIAN' | PDF unavailable |

454 | Motivation for exponential generating function | PDF unavailable |

455 | Recurrrence relation: The theorem and its proof | PDF unavailable |

456 | Introduction to Group Theory | PDF unavailable |

457 | Uniqueness of the identity element | PDF unavailable |

458 | Formal definition of a Group | PDF unavailable |

459 | Groups: Examples and non-examples | PDF unavailable |

460 | Groups: Special Examples Part 1 | PDF unavailable |

461 | Groups: Special Examples Part 2 | PDF unavailable |

462 | Subgroup: Defintion and examples | PDF unavailable |

463 | Lagrange's theorem | PDF unavailable |

464 | Summary. | PDF unavailable |

465 | Conclusion. | PDF unavailable |

Sl.No | Language | Book link |
---|---|---|

1 | English | Not Available |

2 | Bengali | Not Available |

3 | Gujarati | Not Available |

4 | Hindi | Not Available |

5 | Kannada | Not Available |

6 | Malayalam | Not Available |

7 | Marathi | Not Available |

8 | Tamil | Not Available |

9 | Telugu | Not Available |