Question

A-   For all integers a and b, if a + b is odd, then a is odd or b is odd.

B-  For any integer n the number (n³ - n) is even.

C-   Proof of De-Morgan’s Law

Answer 1

A. For all integers $a$  and $b$ let $a+b$ is odd. If $a$ and $b$ are both even than $\exist p, l \in Z a=2p, b=2l\implies a+b=2(p+l)$ is even too. If $a$ and $b$ are both odd than $\exist p, l \in Z a=2p+1, b=2l+1 \implies a+b=2(p+l+1)$ is even too. So we have only one of  $a$  and $b$ is odd and $\exist p, l \in Z a=2p, b=2l+1 \implies a+b=2(p+l)+1$ .

B. For any integer $n$  the number $(n^3-n)$ equals to $n(n+1)(n-1)$ that is the product of three three consecutive natural numbers, one of which is even. In case $(n-1)=0$ we have $(n^3-n)=0$ and it is even too.

C. Let's prove that for alll sets $A$ and $B$ the equality $\lnot(A\land B)=\lnot A\lor \lnot B$ .

$\lnot (A\land B)=(\lnot A\land B)\lor ( A\land \lnot B) \lor ( \lnot A\land \lnot B) =$

$((\lnot A\land B) \lor ( \lnot A\land \lnot B))\lor (( A\land \lnot B) \lor ( \lnot A\land \lnot B)) =$

$( \lnot A \land ( B\lor \lnot B))\lor (\lnot B \land ( B\lor \lnot B)) =\lnot A\lor \lnot B$