MTH202 Final Term Lectures 23-45 by Orange Monkey

MTH202 FOR FINAL TERMA · LECTURE 23 TO 45 · Lecture No.23 Mathematical Induction

What is 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)

Lecture No.24 Mathematical Induction for Divisibility

Type of MATHEMATICAL INDUCTION?

1. DIVISIBILITY PROBLEMS · 2. INEQUALITY PROBLEMS · DIVISIBILITY:

Let n and d be integers and d ≠ 0. Then n is divisible by d or d divides n written as d | n. iff n = d·k for some integer k. alternatively,

We say that n is a multiple of d

D is a divisor of n

D is a factor of n.

Thus d|n⇔∃ an integer k such that n = d·k

Lecture No.25 Methods of proof

What are methods of proof?

A proof is a logical argument that tries to show that a statement is true. There are many different ways to go about proving something,

We’ll discuss 2 methods:

1. Direct proof,

2. INDIRECT PROOF

Type of INDIRECT PROOF

1. Proof by contradiction,
2. Proof by induction.

We'll talk about what each of these proofs are, when and how they're used.

What is DIRECT PROOF?

The implication p →q can be proved by showing that if p is true, the q must also be true. This shows that the combination p true and q false never occurs. A proof of this kind is called a direct proof. p

q · p → q · T · T · T · T · F · F · F · T · T · F · F · T · SOME BASICS: · QUIZ IMPORTANT IN 3quiz and final

1. An integer n is even if, and only if, n = 2k for some integer k.
2. An integer n is odd if, and only if, n = 2k + 1 for some integer k.
3. An integer n is prime if, and only if, n > 1 and for all positive integers r and s, if n = r· s, then r = 1 or s = 1.
4. An integer n > 1 is composite if, and only if, n = r· s for some positive integers r and s with r ≠ 1 and s ≠ 1.
5. A real number r is rational if, and only if, a/ b for some integers a and b with b≠0.
6. If n and d are integers and d ≠0, then d divides n, written d | n if, and only if, n = d.k for some integers k.
7. An integer n is called a perfect square if, and only if, n = k2 for some integer k.

Question

imp

Assume that m and n are particular integers. Justify your answer to each of the following:

Q1. Is 4 6 m n + even?

Q2. Is 8 5 mn + odd?

Solution:

1. Yes: 4m+6n =2.(2m+3n) [by definition of even] (2m+3n) is an integer because 2, 3, m, n are integers and products and sums of integers are integers.
2. Yes: 8mn+5 =2.(4mn+2)+1 [by definition of odd] (4mn+2) is an integer because 4, 2, m, n are integers and products and sums of integers are integers.

Question

imp

Prove that if x divides y and y divides z then x divides z.