Please answer and show your work. Thumbs up guaranteed. Thank you Please provide
ID: 3195751 • Letter: P
Question
Please answer and show your work. Thumbs up guaranteed. Thank you
Please provide a proof. Please be concise and use proper notation, thank you.
If n2+1 is even, then n2 is odd.
Is this attempt valid?
Proof by contraposition. If n2 is even, then n2+1 is odd.
Let's assume n2 is even, then n2 = 2k for some integer k.
Consequently, n2+1 = 2k+1
This shows that n2+1 is odd since it is of the form 2k+1. Therefore if n2+1 is even, then n2 is odd.
If you have another method please show me and let me know if my attempt is a valid one!
Explanation / Answer
The attemt is a valid one.
The other way od proving it is proof by contradiction.
Show that if the statement is F, it leads to a contradiction. The statement is F when n^2 + 1 is even, and n^2 is even, so assume n^2 +1 is even, and n^2 is even.
Since, n^2 is even it can be written in the form of 2k
=> n^2 + 1 = 2k + 1
=> n^2 + 1 is odd as it can be expessed in the terms of 2k+1
This contradicts our assumption. Thus the statement is proved by contradiction