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I need to write a proof by contradiction to show that if $x^2 +2x-3=0$ then $x\neq 2$

To do a proof by contradiction, you assume the opposite. So I would assume that if $x^2+2x-2$ is not equal to $0$, then $x=2$.

To prove: Assume $x^2+2x-2$ is not equal to $0$ for $x$ in $E$....

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What you say is not a proof by contradiction but a logical mistake: you want to prove $\,A\implies B\,$ by means of $\,\neg A\implies \neg B\;$ , which is wrong. What you can do is the counterpositive $\,\neg B\implies \neg A\;$ ... –  DonAntonio Jul 1 '13 at 0:34

4 Answers 4

First: To prove $\;p\implies q$ using a proof by contradiction, it is not a good idea to think of it as "assuming the opposite." Assuming the opposite of what? In logic as in math, it's good to think of "opposite" of an assertion as its negation. And importantly, you need to remember here exactly which part of the overall assertion gets "negated" in our assumption:

We have an overall assertion which we call an implication, which has the form $\;p \implies q$: $\;p$ happens to also be an assertion (serving as a premise, or antecedent, of the implication) which is said to imply another assertion, $q$ (which we call the conclusion, or consequent, of the implication.)

Now: To prove $p \implies q$ using a proof by contradiction, you assume the premise $p$, and you also assume the negation of $q$, that is, you assume the negation of the conclusion.

In this case, you start by assuming that the premise $x^2 + 2x - 3 = 0$ is true, and by assuming that it is not the case that the conclusion "$x \neq 2$" holds(i.e., you assume it is false that $x \neq 2)$. In other words, here, we assume $x^2 + 2x - 3 = 0\;$ and suppose $\bf x = 2$. Now the aim is to arrive at a contradiction.

.....[Work needed: find the contradiction].....

Then, having found a contradiction, you can conclude that the assumption $(x = 2)$ is false, and therefore, it follows that if the premise $x^2 + 2x - 3$ is true, then the conclusion $\lnot (x = 2) \iff \;x \neq 2$ must also be true.

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so to find the contradiction, I would do (x^2) +2x -2=0 for x+2. Then (2^2) + 2(2)-2=0 which is not true. Is this correct? –  sophie Jun 30 '13 at 23:48
oops, I think you mean $-3$ in the equation. But yes, Exactly! substitute, as you did, evaluate, and Then you can reject the "supposition that $x = 2$" and claim that if $x^2 + 2x - 3 = 0$, then $x\neq 2$. –  amWhy Jun 30 '13 at 23:49
In either case, if we have $x^2 + 2x - 3 = 0$ or $x^2 + 2x - 2 =0$, you can see that we get a contradiction if we suppose $x = 2.$ In either case, by arriving at a contradiction, you can conclude that $x\neq 2$ –  amWhy Jun 30 '13 at 23:57
This helps so much!!! Thank you!!! –  sophie Jul 1 '13 at 2:36
You're welcome, sophie! –  amWhy Jul 1 '13 at 2:37

You took the revese meaning of contradiction.

If you have to prove anything by contradiction it means that the answer which you have to prove is false(x is not equal to 2) and not the statement which you will use to prove the same ($x^2+2x-3=0$).

The basic idea behind contradiction is that you assume that the thing you need to prove is not correct and you $contradict$ yourself by proving that you are wrong, that everything you've assumed is wrong and since you just assume the opposite, the opposite of the opposite i.e. the thing you assumed wrong is correct.

This is an analytical approach to try your questions. If there is a general statement $\mathbb x$ and from it you have to prove statement $\mathbb a$ then you take the converse of it {$\mathbb b$} to do the job. You put your assumption back to $\mathbb x$ and if it follows then your answer is $\mathbb b$ else it is $\mathbb a$.

So here, $$x^2+2x-3=0$$ We assume$x=2$. So we get $5=0$. So our assumption is wrong.

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When you try to prove a logical proposition by contradiction, you have to assume the opposite. In your case $p\Longrightarrow q$ is false only when $p$ is true and $q$ is false, translating this to your problem:

If $x^2+2x-3=0$, then $x\neq 2$.

We assume $p$ is true: $x^2+2x-3=0$.

And $q$ is false: $x=2$.

Are those two things compatible, can you see the contradiction here?

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You were probably thrown by the ridiculously easy nature of the answer, which is this:

If $x$ were equal to $2$, then $x^2+2x-3$ would equal $5$, not $0$.

Hence (duh!) $x$ is not equal to $2$.

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