a. All programmers enjoy discrete mathematics
Negation:There is exists at leat one programmer who doesn't enjoy discrete mathematics
~$\forall$x[P(x)--> L(x)] = ~$\forall$x[~P(x) V L(x)] = $\exists$x[P(x) AND ~L(X)]
b. Some integers are not odd.
Negation: All integers are odd.
~[$\exists$x~O(x)] = $\forall$O(x)
c. Every integer that is divisible by 2 is even.
Negation: There is at least one integer that is divisible by 2 and is not even.
~[$\forall$x( $\exists$D[x=2D]-->E(x)] = ~[$\forall$x($\forall$D[x!=2D] V E(x))] = $\exists$x[$\exists$D[x=2D] AND ~E(X)]
d. There exists a natural number that is not a positive integer
Negation: All natural numbers are positive integer.
~[$\exists$x(N(x) AND ~P(x))] = $\forall$x[~N(x) V P(x)) = $\forall$x[N(x) --> P(x)]