Lecture 008

Conditional Claims

Claim: $(\forall n \in \mathbb{Z})((\exists l \in \mathbb{Z}(n^2 = 2l) \implies (\exists m \in \mathbb{Z})(n=2m)))$ (if n^2 is even, then n is even) Proof: since assuming n^2 is even is hard, we prove by showing "if n is not even, then n^2 is not even". (logically equivalent)

We want to start from the left hand side, if left hand side is not simple, we can use contrapositive

Claim: a rational number added with a irrational number is irrational Proof: AFSOC, define rational and irrational using sets. Then show x+y can be represented by fractions, then isolate the irrational y by doing y=(x+y)-x, that it is fractional.

Biconditional Claims

Two techniques:

1. (typical method) prove $P \implies Q$ and $Q \implies P$
2. create a chain of logical equivalences ($P \equiv A \equiv B \equiv C \equiv Q$)

WLOG: without loss of generality. when two statement only differ by name of the variable.

WTS: want to show

Basic Set Theoretic Proofs

More likely to use chains of logical equivalences

Claim: For any sets A and B, $\mathcal{P}(A) \cap \mathcal{P}(B) = \mathcal{P}(A \cap B)$ Proof: let U be a universal set containing $\mathcal{P}(A)$ and $\mathcal{P}(B)$, and let $X \in U$ be arbitary, then

• $X \in \mathcal(A) \cap \mathcal{B} \equiv X \in \mathcal{A} \land X \in \mathcal{B}$

• $\equiv X \subseteq A \land X \subseteq B$

• $\equiv X \subset A \cap B$

• $\equiv X \in \mathcal{A \cap B}$

Claim: $(\forall x, y \in \mathbb{R}(x^3 + x^2y = y^2+xy) \equiv (y=x^2 \lor y=-x))$ Proof: Let $x,y \in \mathbb{R}$ be arbitary and fixed. then:

• $x^3+x^2y = y^2+xy \equiv x^3+x^2y-y^2-xy = 0$

• $\equiv (x^2-y)(x+y) = 0$

• $\equiv (y=x^2)\lor (y=-x)$