Understanding requirement of logic question

Question

I have a statement like this: if 4x - 3 is even then x is odd. What does providing the predicate for the starting assumption and the predicate for the concluding assumption for a proof by contraposition mean?

Current I have some like this as the predicate for the starting assumption:

P(x) = 4x - 3 is odd

Is this sufficient? or do I need to go further.

Answer 1

Given that you have to prove $P \implies Q$ for two propositions $P$ and $Q$, the contraposition asks you to prove the equivalent statement $\neg Q \implies \neg P$, where $\neg$ stands for negation.

Therefore, for attempting a proof by contraposition, the starting assumption is $\neg Q$, where $Q$ is the conclusion of the original statement.

Similarly, the conclusion of a proof by contraposition, is $\neg P$, where $P$ is the starting assumption of the original statement.

In our case, we have $P = 4x-3$ is even and $Q = x$ is odd.

Hence, $\neg Q = \mathit{x}$ is even and $\neg P = \mathit{4x-3}$ is odd are the starting assumption and conclusion of a proof by contraposition, respectively.

Therefore, for attempting a proof by contraposition, you start with the fact that $x$ is even. You should now show that $4x-3$ is odd, which is the conclusion,using the starting assumption.

Note : In our case, it turns out the conclusion is true if $x$ is any integer, so starting assumption is somewhat stronger than required.

Answer 2

As we know, to prove P -> Q, you have to ensure that there is no situation in which P is false and Q is true.

Based on this statement, your questions looks like this:

Even(4x - 3) -> Odd(x)

where, Even(x): x is even and Odd(x) : x is odd

To prove the statement by, Proof by contraposition, you need to know that

P -> Q ≡ ~Q -> ~P, where ~a is negation of a. -------- (I)

So, you need to assume about ~Q or more specifically ~Odd(x).

Assume, Odd(x) is false, meaning x is even.

x = 2k, for any arbitrary Integer k.

Then 4x - 3 = 4(2k) - 3 = 8k - 3.

We know 8k is Even, and 3 is odd. Also Even - Odd = Odd.

This means Even(4x - 3) is also false.

As we proved it's contrapositive, we know by (I) that the claim is also true.

QED.