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

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:

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