(1) Prove, using propositional calculus, that
~((PvQ)>R) <=> (P^~R)v(Q^~R)
(2) Given an example of a conjunction that is a tautology and a disjunction that is a contradiction
(3) Negate and put into positive form
a: (?x)[Q(x)>P(x)]
b: (∀x)(?y)(P(x)^Q(x))
c: (?x)(?y)(P(x)vQ(x,y))
d: [(?x)(~R(x))]v[(∀x)(Q(x)<>~P(x))]
(4) Let x and y be two integers. Prove that x^2+xy+y^2 is divisible by 9 if and only if x and y are divisible by 3. (Please provide the (=>) proof only, proof by contradiction)