Please help with negation!!!

Question

The negation of the sentence "If the product   is even, then  is even or  is even" is?

Answer 1

Hi Fizaa,

If the product xy is NOT even, then x is NOT even AND y is NOT even.

This is true by the associative property of negation.

BR,

Omar

Answer 2

One way of doing this is to notice that (a→b) is logically equivalent to (¬a ∨ b)

Then ¬(a→b) is equivalent to ¬(¬a ∨ b) which is equivalent to ¬¬a ∧ ¬b or a ∧ ¬b.

Here a is the statement that "x*y is even"; b is the statement (x is even ∨ y is even) where ∨ is or

The negation will be a ∧ ¬b or (x*y is even) ∧ ¬((x is even) ∨ (y is even))

We can simplify this expression to (x*y is even) ∧ (¬(x is even) ∧ ¬(y is even))

Which simplifies to (x*y is even) ∧ ((x is odd) ∧ (y is odd))

So one form of the negation is "The product x*y is even and x is odd and y is odd". This statement is of course FALSE because an odd*odd is always odd. We negated a statement that is always true, so it is logical that we should arrive at this conclusion.