The problem involves determining the contrapositive of a logical statement regarding the evenness of two integers, x and y, and their product and sum. The original statement asserts that if both the product (xy) and the sum (x+y) are even, then both integers x and y must also be even.
Multiple interpretations of the contrapositive are being discussed, with some participants affirming the validity of certain formulations while others express confusion about the logical connections. Guidance has been offered regarding the negation of the original statement.
There is a focus on the logical implications of the original statement, with participants questioning the correct interpretation of "not even" and its relation to odd numbers. The discussion reflects a mix of understanding and uncertainty regarding logical constructs.
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.bonfire09 said: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?
To whom are you responding?bonfire09 said: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.