Lecture 009

Biconditional Claims

Claim: for any a and b, a and b have the same parity (they are all even or all odd) iff there exists m,n \in \mathbb{Z}(a=m+n \land b=m-n)

(\forall a,b \in \mathbb{Z})((\exists k, l \in \mathbb{Z})((a=2k \land b=2l)\lor (a=2k+1 \land b=2l+1)) \equiv (\exists m,n \in \mathbb{Z})(a-m+n \land b=m-n))


Claim: for all integers m and n m^2(n^2-2n) is odd iff m and n are both odd. Proof: using definition only. easy, same as above

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