# What would be the contrapositive of this statement?

## Homework Statement

The original statement is Prove that if xy and x+y are even then both x and y are even.

## The Attempt at a Solution

I think it goes like "If x or y is odd then xy and x+y are odd"? I'm not too sure though because the first "and" in the hypothesis is confusing. I'm not sure if its supposed to go like"If x or y is odd then xy or x+y is odd?

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jbunniii
Homework Helper
Gold Member

## Homework Statement

The original statement is Prove that if xy and x+y are even then both x and y are even.

## The Attempt at a Solution

I think it goes like "If x or y is odd then xy and x+y are odd"? I'm not too sure though because the first "and" in the hypothesis is confusing. I'm not sure if its supposed to go like"If x or y is odd then xy or x+y is odd?
You want to negate the statement "xy and x+y are even". If this statement is not true, it means that either xy is not even or x+y is not even, or both. And "not even" is the same as "odd". So indeed "If x or y is odd then xy or x+y is odd" is correct.

HallsofIvy
Homework Helper
The statement is "If ((x is even) and (y is even)) then ((xy is even) and (x+y is even))".

As you suggest, the contrapositive of "if A then B" is "if not B then not A". So the contrapositive of this statement is
"if NOT ((xy is even) and (x+y is even)) then NOT ((x is even) and (y is even))".

The thing you are missing is that "NOT A and B" is "(Not A) or (not B)".
So the contra positive is
"if xy is odd or x+y is odd, then either x is odd or y is odd".

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I would think the sentence would have to be " If x is odd or y is odd then xy is odd or x+y is odd." I think you have it flipped around.

SammyS
Staff Emeritus