Thread: Can anyone help me with 2 homework questions?

Been stuck on these two for a while now. Would anyone be able to help me with them? Thanks!

#1.) Prove by contradiction that for all n ∈ Z, if n^3 + 5 is odd, then n is even

#2.) Prove using a direct proof that for every rational number x, if x ≠ 0, then 1/x is rational

1)
Assume n is not even. Set n=2k+1 where k is a natural number
Show that the assumption is false because n^3 +5 cannot be even when n=2k+1.

2)
Let
Show that 1/x can be written as a quotient.

Originally Posted by Shakarri

1)
Assume n is not even. Set n=2k+1 where k is a natural number
Show that the assumption is false because n^3 +5 cannot be even when n=2k+1.

2)
Let
Show that 1/x can be written as a quotient.

Thank you!!

Originally Posted by mgk501

Been stuck on these two for a while now. Would anyone be able to help me with them? Thanks!

#1.) Prove by contradiction that for all n ∈ Z, if n^3 + 5 is odd, then n is even

#2.) Prove using a direct proof that for every rational number x, if x ≠ 0, then 1/x is rational

It's easier to prove the first question using the contrapositive. Your statement is equivalent to saying if n is odd, then is even. If n is odd, we can write it as n = 2m + 1. So



which is clearly even. Q.E.D.

"Proving the contrapositive" is a proper subset of "proof by contradiction".

Originally Posted by HallsofIvy

"Proving the contrapositive" is a proper subset of "proof by contradiction".

I don't see how that could possibly be. In my example I did NOT reach a contradiction...