Statement S: If the sum of two integers is even, then both the difference of the two integers is even and the product of the two numbers is even.
a. Rewrite the statement S with no words, using only mathematical symbols and variables. (Do not use open sentences.) Hint: This should begin with a quantifier.
b. Write the negation of S, maximally negated, using only mathematical symbols and variables.
c. Write the converse of S using only mathematical symbols and variables.
d. Write the contrapositive of S using only mathematical symbols and variables.
e. Prove or disprove the statement S
My attempt
a. $\forall$$x,y$$\in$$\mathbb{Z}$, $[2\mid(x+y)]$ $\Rightarrow$ $([2\mid|x-y|]$ $\wedge$ $[2\mid xy ])$
b. $\neg$$\forall$$x,y$$\in$$\mathbb{Z}$,$[2\mid(x+y)]$ $\Rightarrow$ $([2\mid|x-y|]$ $\wedge$ $[2\mid xy ])$
Move past quantifier: $\exists$$x,y$$\in$$\mathbb{Z}$, $\neg$($[2\mid(x+y)]$ $\Rightarrow$ $([2\mid|x-y|]$ $\wedge$ $[2\mid xy ])$)
Negate implication: $\exists$$x,y$$\in$$\mathbb{Z}$, $[2\mid(x+y)]$ $\wedge$ $\neg$$([2\mid|x-y|]$ $\wedge$ $[2\mid xy ])$
By Demorgan's law: $\exists$$x,y$$\in$$\mathbb{Z}$, $[2\mid(x+y)]$ $\wedge$ $([2\nmid |x-y|]$ $\lor$ $[2\nmid xy ])$
c. $\forall$$x,y$$\in$$\mathbb{Z}$, $([2\mid|x-y|]$ $\wedge$ $[2\mid xy ])$ $\Rightarrow$ $[2\mid(x+y)]$
d. $\forall$$x,y$$\in$$\mathbb{Z}$, $([2\nmid|x-y|]$ $\lor$ $[2\nmid xy ])$ $\Rightarrow$ $[2\nmid(x+y)]$
e. Counterexample: Let $x=3$, $y=1$. Then $2$$\mid$$4$ but $2$$\nmid$$3$; that is, if $x=3$ and $y=1$, then $2$$\mid$$(x+y)$ but $2$$\nmid$$xy$.