What would be the contrapositive of this statement?

In summary, a contrapositive statement is a logical statement formed by switching the hypothesis and conclusion of a conditional statement and negating both. To write a contrapositive statement, the given conditional statement is identified and the hypothesis and conclusion are switched with the word "not" added in front of both parts. The purpose of finding the contrapositive of a statement is to prove its logical equivalence and simplify complex statements. Both a statement and its contrapositive can be true since they have the same truth value. The contrapositive of a statement can also be used in mathematical proofs, particularly in proofs by contradiction, to show the truth of a statement by proving its contrapositive is true.
  • #1
bonfire09
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0

Homework Statement


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


Homework Equations





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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  • #2
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?
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.
 
  • #3
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".
 
  • #4
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  • #5
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.
 
  • #6
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.
To whom are you responding?

(We have a "QUOTE" feature to take the guess-work out of this.)

By the Way: " If x is odd or y is odd then xy is odd or x+y is odd." is a valid contrapositive to the original statement.
 

1. What is the definition of a contrapositive statement?

A contrapositive statement is a logical statement that is formed by switching the hypothesis and conclusion of a conditional statement and negating both.

2. How do you write a contrapositive statement?

To write a contrapositive statement, first identify the given conditional statement. Then, switch the hypothesis and conclusion of the statement and add the word "not" in front of both parts.

3. What is the purpose of finding the contrapositive of a statement?

The purpose of finding the contrapositive of a statement is to prove the original statement is logically equivalent to its contrapositive. This can help in simplifying complex statements and making logical arguments.

4. Can a statement and its contrapositive both be true?

Yes, a statement and its contrapositive can both be true. This is because they are logically equivalent and have the same truth value. If the original statement is true, the contrapositive will also be true.

5. How can the contrapositive of a statement be used in mathematical proofs?

The contrapositive of a statement can be used in mathematical proofs to show that a statement is true by proving its contrapositive is true. This is often used in proofs by contradiction, where the negation of a statement is assumed to be true and a contradiction is found, proving the original statement to be true.

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