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Table of Contents

LOGIC ..............................................................................................................................3 TRUTH TABLES.............................................................................................................9 LAWS OF LOGIC..........................................................................................................17 BICONDITIONAL.........................................................................................................24 ARGUMENT..................................................................................................................29 APPLICATIONS OF LOGIC ........................................................................................34 SET THEORY................................................................................................................41 VENN DIAGRAM.........................................................................................................46 SET IDENTITIES ..........................................................................................................58 LECTURE NO.10 APPLICATIONS OF VENN DIAGRAM.....................................................................65 LECTURE NO.11 RELATIONS ..................................................................................................................73 LECTURE NO.12 TYPES OF RELATIONS...............................................................................................80 LECTURE NO.13 MATRIX REPRESENTATION OF RELATIONS ......................................................90 LECTURE NO.14 INVERSE OF RELATIONS..........................................................................................97 LECTURE NO.15 FUNCTIONS................................................................................................................102 LECTURE NO.16 TYPES OF FUNCTIONS ............................................................................................112 LECTURE NO.17 INVERSE FUNCTION................................................................................................123 LECTURE NO.18 COMPOSITION OF FUNCTIONS.............................................................................134 LECTURE NO.19 SEQUENCE..................................................................................................................144 LECTURE NO.20 SERIES .........................................................................................................................151 LECTURE NO.21 RECURSION I .............................................................................................................159 LECTURE NO.22 RECURSION II............................................................................................................165 LECTURE NO.23 MATHEMATICAL INDUCTION ..............................................................................170 LECTURE NO.24 MATHEMATICAL INDUCTION FOR DIVISIBILITY...........................................179 LECTURE NO.25 METHODS OF PROOF...............................................................................................186 LECTURE NO.26 PROOF BY CONTRADICTION.................................................................................193 LECTURE NO.27 ALGORITHM ..............................................................................................................201 LECTURE NO.28 DIVISION ALGORITHM ...........................................................................................205 LECTURE NO.29 COMBINATORICS.....................................................................................................210 LECTURE NO.30 PERMUTATIONS .......................................................................................................218 LECTURE NO.31 COMBINATIONS........................................................................................................226 LECTURE NO.32 K-COMBINATIONS ...................................................................................................232 LECTURE NO.33 TREE DIAGRAM ........................................................................................................238 LECTURE NO.34 INCLUSION-EXCLUSION PRINCIPLE...................................................................245 LECTURE NO.35 PROBABILITY............................................................................................................250 LECTURE NO.36 LAWS OF PROBABILITY .........................................................................................256 LECTURE NO. 37 CONDITIONAL PROBABILITY..............................................................................265 LECTURE NO. 38 RANDOM VARIABLE..............................................................................................272 LECTURE NO. 39 INTRODUCTION TO GRAPHS ...............................................................................280 LECTURE NO. 40 PATHS AND CIRCUITS ...........................................................................................289 LECTURE NO. 41 MATRIX REPRESENTATION OF GRAPHS..........................................................296 LECTURE NO. 42 ISOMORPHISM OF GRAPHS...................................................................................303 LECTURE NO. 43 PLANAR GRAPHS ....................................................................................................312 LECTURE NO. 44 TREES .........................................................................................................................319 LECTURE NO. 45 SPANNING TREES....................................................................................................327

Mathematical Induction

PRINCIPLE OF MATHEMATICAL INDUCTION:

Let P(n) be a propositional function defined for all positive integers n. P(n) is true for every positive integer n if

1.Basis Step:

The proposition P(1) is true.

2.Inductive Step:

If P(k) is true then P(k + 1) is true for all integers k ≥ 1. i.e. ∀ k p(k) → P(k + 1)

EXAMPLE:

SOLUTION:

Let

1.Basis Step:

P(1) is true. For n = 1, left hand side of P(1) is the sum of all the successive integers starting at 1 and ending at 1, so LHS = 1 and RHS is

so the proposition is true for n = 1.

2. Inductive Step: Suppose P(k) is true for, some integers k ≥ 1.

(1)

(2) 1) n n n

1) ( ) :1 n n P n n

1(1 1) R H S

1) k k k

1)( 2) 1) k k k

1) 1)

1) 1) using (1)

1) k k k

k k k

k k