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.
Two techniques:
WLOG
: without loss of generality. when two statement only differ by name of the variable.
WTS
: want to show
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)
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