Negate: If 15 | x, then 3 | x and 5 | x.

Answer 1

Let's denote the proposition "15|x" by p and "3|x and 5|x" by q. Then we're required to negate p --> q, that is, "if p, then q". We can easily check (by a truth table, for example) that p --> q is equivalent to "~p or q" (read "not p or q"). Hence, the required negation is ~(~p or q), which by De Morgan's laws, is equivalent to ~~p and ~q, that's to say, p and ~q. Hence, the negation in plain English is: 15|x and not (3|x and 5|x). With another application of De Morgan's laws, this becomes: 15|x and (3 doesn't divide x or 5 doesn't divide x).