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I am trying to determine whether an argument is valid.

The question reads: "If $x^2 \neq 0$, where $x$ is a real number, then $x \neq 0$. Let $a$ be a real number with $a^2 \neq 0$; then $a \neq 0$".

First of all I am confused by the two statements here. Are they not both saying the same thing, just with different variables?

Since there are two statements like this, how do I properly prove this is valid, or alternatively, not valid?

ComputerLocus
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2 Answers2

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The statements are indeed valid. I believe the point of the question is to make you realize that the use of the letter $a$ as opposed to $x$ has no impact on the statement.

nullUser
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  • So what would I be doing to prove this then? Would I be proving the $x$ variable version and then just saying the proof can be applied to $a$? – ComputerLocus Feb 06 '15 at 04:22
  • I agree that must be the point of the question, but what a stupid question. This is why we can't have nice things. – Daniel W. Farlow Feb 06 '15 at 04:26
  • The lesson is not stupid. A ridiculous number of students think that saying $f(x) = g(x)$ is different from saying $f(t)=g(t)$. – nullUser Feb 06 '15 at 04:28
  • @nullUser If that's true, then...what kind of teaching has been circulating? That makes me sad. – Daniel W. Farlow Feb 06 '15 at 04:29
  • NullUser would you advise solving this by contraposition? The contraposition concept is still fairly new to me and I am a little confused with exactly how you setup things when working with contraposition? – ComputerLocus Feb 06 '15 at 04:29
  • It depends on what the questions is. Is the question "determine whether this argument is valid"? Or is the question to actually prove that if $x^2 \neq 0$ then $x \neq 0$ using axioms of $\mathbb{R}$? – nullUser Feb 06 '15 at 04:30
  • It is to determine whether or not the argument is valid. So I assume if I am going to say it is valid I would need to have a proof showing that it is a valid argument. – ComputerLocus Feb 06 '15 at 04:33
  • "An argument is valid if and only if the truth of its premises entails the truth of its conclusion." The premises of the given argument are $x\in \mathbb{R}\wedge x^2 \neq 0 \implies x \neq 0$. Indeed, from this premise, if $a \in \mathbb{R}\wedge a^2 \neq 0$ we may conclude that $a \neq 0$. – nullUser Feb 06 '15 at 04:36
  • So in doing it this way you would be assuming that the premise is true, and therefore can apply it to $a$? – ComputerLocus Feb 06 '15 at 04:49
  • Yes. Validity of an argument does not check the truth of the premises. For an argument to be valid, you get to assume the premises, and you only must check that given the premises you logically arrive at the conclusion. An argument is called sound if it is valid and all of its premises are true. – nullUser Feb 06 '15 at 04:51
  • And what if the teacher intended us to have to prove $x^2 \neq 0$ and $x = 0$ before we applied it to $a^2$'s? Would I have to first then do something like what Andrew Farber showed in his answer with contrapositive proof, to prove the statement, and then add an explanation like you gave to apply it to $a^2$? – ComputerLocus Feb 06 '15 at 04:54
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    The teacher did not intend you to prove $x^2 \neq 0 \implies x \neq 0$ because he or she asked you if the argument is valid. If you wanted to prove $x^2 \neq 0 \implies x \neq 0$ the contrapositive works fine though. – nullUser Feb 06 '15 at 04:57
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The argument is valid. Using the contrapositive:

Suppose $x \in \mathbb{R}$ and $x = 0$. Then $x^2 = x \times x = 0 \times 0 = 0$. Let x = a.

Andrew
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