(Direct proof) A claim is given as a quantified statement: “The product of two odd numbers is an odd number” a) (1 point) Write the domain of the variables: b) (4 points) Write the statement using quantifiers and an implication of propositional functions: c) (15 points) Prove the statement by direct proof (Assume the hypothesis and derive the conclusion)

Let $P(x)$ denotes the statement "$x$ is an odd number", $Q(x,y)$ denotes the statement "$x\cdot y$ is an odd number".

a) It follows that the the domain of the variables is the set $\Z$ of integers.

b) Let us write the statement using quantifiers and an implication of propositional functions: $\forall (x\in\Z)\forall(y\in\Z)\ (P(x)\land P(y)\to Q(x,y))$

c) Let us prove the statement by direct proof. Let $x\in \Z$ and $y\in\Z$ be odd integers. Then $x=2k+1,y=2t+1$ for some $k,t\in\Z.$ It follows that $x\cdot y=(2k+1)\cdot(2t+1)=4kt+2k+2t+1=2(2kt+k+t)+1$, and hence $x\cdot y$ is an odd number.