Introduction to Real Analysis

Mathematical logic

Proposition

A statement in either true or false state.

Symbols

Symbol	Read as
	and
	or
	then
	implies
	implied by
	if and only if
	for all
	there exists
	not

Let’s take .

1. Contrapositive or transposition: . This is equivalent to the original.
2. Inverse: . Does not depend on the original.
3. Converse: . Does not depend on the original.

Examples

Methods of proofs

1. Just proof what should be proven
2. Prove the contrapositive
3. Proof by contradiction
4. Proof by induction

Proof by contradiction

Suppose has to be proven. If is proven to be false, then, by proof by contradiction, can be trivially proven.

Logic behind proof by contradiction