Proof verification for proof by contrapositionprove using contraposition that if a and b are integers:...

racodelitusmn

racodelitusmn

Answered

2022-07-12

Proof verification for proof by contraposition
prove using contraposition that if a and b are integers:
a 2 4 b 2 0
So the contrapositive:
a 2 4 b 2 = 0 implies that at least one of either a or b are not integers
Working from the above, I got: a 2 = 4 b + 2
a 2 = 2 ( 2 b + 1 )
If both a and b are integers, 2 ( 2 b + 1 ) is a perfect square.
In other words: 2 ( 2 b + 1 ) = k
Where k is an integer
I also realised that 2 b + 1 is an odd number so:
k = 2 2 b + 1
I ended my proof by saying that since 2 b + 1 is odd, it does not have a multiple of 2 so k will always have multiple or 2 which means that it cannot be an integer.

Answer & Explanation

pompatzz8

pompatzz8

Expert

2022-07-13Added 11 answers

Explanation:
You're almost done. You can conclude your proof as follows:
a 2 = 2 ( 2 b + 1 ) immediately implies, you have a = 2 k , k Z . Therefore, we have 4 k 2 = 2 ( 2 b + 1 ) 2 k 2 = 2 b + 1 which gives a contradiction.

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