Can Proof by Contradiction Be Validated through Logical Equivalences?

[Cole A.]

Homework Statement

Prove that if x^2 + y = 13 and y \neq 4, then x \neq 3.

Homework Equations

N.A.

The Attempt at a Solution

The proof itself is simple enough: suppose x^2 + y = 13 and y \neq 4. Suppose for the sake of contradiction that x = 3. Then
<br /> \begin{align*}<br /> (3)^2 + y &amp;= 13 \\<br /> y &amp;= 4.<br /> \end{align*}
But this contradicts the knowledge that y \neq 4. Therefore, if x^2 + y = 13 and y \neq 4, then x \neq 3.

The problem I am having is understanding why this is logically valid. Would it be correct to say that, for the statements
<br /> \begin{align*}<br /> A &amp;: x^2 + y = 13 \\<br /> B &amp;: y = 4 \\<br /> C &amp;: x = 3,<br /> \end{align*}<br />
what has been proven is below?
<br /> \begin{align*}<br /> &amp;(A \wedge \neg B \wedge C) \rightarrow B \\<br /> \text{which is equivalent to}~ &amp;(A \wedge \neg B) \rightarrow (C \rightarrow B) \\<br /> \text{which is equivalent to}~ &amp;(A \wedge \neg B) \rightarrow (\neg B \rightarrow \neg C) \\<br /> \text{which is equivalent to}~ &amp;(A \wedge \neg B \wedge \neg B) \rightarrow \neg C \\<br /> \text{which is equivalent to}~ &amp;(A \wedge \neg B) \rightarrow \neg C.<br /> \end{align*}<br />

Is this the proper way to think about the validity of proof by contradiction? (Sorry if this is a dumb question, I'm not a mathematician. What I am finding hard to stomach is identifying x = 3 as the contradictory statement when there are actually three statements that were assumed to be true (and thus possible culprits of the contradiction)).

 

[Infrared]

There is no need to write out the argument in symbolic logic. It works as follows: If x were 3 then y would have to be 4. Since we know that y is not 4, our assumption that x=3 must be false since we obtained an incorrect result with it. Therefore x \neq 4.

You ask why the other assumptions couldn't be the source of the contradiction. We are taking x^2 + y = 17 and y \neq 4 for granted. After all, the goal of the proof is to show that \mathbf {if} these two statements are true, then x \neq 4.

 

[SammyS]

... then x \neq 4.

I'm sure that you have a typo.

You mean \ x \neq 3 .

 

[Cole A.]

@HS-Scientist: Thanks for your answer.