Lecture 006

Proof Writing

Direct Proofs: attempt to prove statement is itself true Indirect Proofs: prove the negation of the statement is false. (proofs by contradiction: AFSOC - assume for the sake of contradiction)

Universal Statements

Universal Statement: $(\forall x \in S)P(x)$

• Direct Proofs: Let $x\in S$ be arbitrary and fixed. Show $P(x)$ holds.

• Claim: $(\forall x,y \in \mathbb{R})(4x^6+2y^2 \geq 4x^3 y)$

• Proofs: in English

Set Containment Proofs: show $A \subset B$ by showing $(\forall a \in A)(a\in B)$

• Let... be arbitrary and fixed, then [extract some property of fixed arbitrary value]. Since [property], and... is arbitrary, we conclude [result].

Double Containment Proof:

• $A\subset B$, $B\subset A$, $A=B$

(see example 6.3 on lecture note Here)

QED($\blacksquare$): quod erat demonstrandum (proof is done)