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I have a question about how to interpret arbitrary elements.

When we say:

  • "let $x$ be an arbitrary integer"

  • "let $x$ be any integer"

  • "let $x$ be an integer"

at the beginning of proofs, are we saying:

  • "let the symbol $x$ represent one of the integers, although it is not clear which specific integer the symbol $x$ represents".

If not, please let me know how I should think about the above statements.

Thank you for your time.

Edit:

I had a follow up question to the one I asked earlier. What do mathematicians mean by “arbitrary element” or “any element”. I can understand when mathematicians say “let $x$ be $4$” as this means “the symbol $x$ represents $4$”. Could someone please elaborate on why the specific terms “arbitrary” and “any” are used?

Pranav Jain
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    I would think of it more as "let the symbol $x$ represent one of the integers, where we are intentionally not specifying a particular value of $x$ so that any conclusions we draw regarding $x$ will hold for all integers." – DMcMor Dec 18 '20 at 21:33
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    @DMcMor I just wanted to ask for one more clarification. Is what you said above "let the symbol $x$ ... hold for all integers" how mathematicians actually think whenever they read statements like "let $x$ be any integer" or "let $x$ be an arbitrary integer". It seems as though, the way the statements are worded, that mathematicians are trying to invoke a specific idea, like that of randomly choosing an integer, and using the symbol x to represent that integer. – Pranav Jain Dec 19 '20 at 00:23
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    It's not random, and I wouldn't call it unknown. It's arbitrary. That is, it could stand in for any integer. It's as if you said "Suppose a customer walks into the store. Then you must greet that customer." In the second sentence the word customer can stand in for any customer. It's not that it's a mystery customer, or a random customer. It's a trick of language that lets us talk about any customer without other assumptions about them. – Jair Taylor Dec 19 '20 at 00:54
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    @JairTaylor Ok, so when we say "Let $x$ be an arbitrary integer", the symbol $x$ has the meaning "a integer" or "a generic integer"? – Pranav Jain Dec 19 '20 at 01:06
  • @PranavJain Sure. – Jair Taylor Dec 19 '20 at 01:07
  • I would say that the sentence “let $x$ be an arbitrary integer” is equivalent to placing everything that follows inside the scope of a “$\forall x\in \mathbb{Z}$” universal quantifier. – mjqxxxx Dec 19 '20 at 03:37
  • @mjqxxxx That is quite an interesting line of reasoning I have not thought of before. Could you maybe provide a short example? – Pranav Jain Dec 19 '20 at 03:39
  • You should read this post. – user21820 Dec 30 '20 at 12:20

1 Answers1

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Your statement is somewhat correct, but maybe not that useful.

Consider a statement like the following, which we'll call Statement 1:

Let $x$ be an integer. The number $4x+3$ is odd.

The power of this statement is its universality: it applies to every integer $x$. If I wanted, I could say Statement 2:

If $x=-26$, then $4x+3$ is odd.

This is a less general statement that only applies to a single integer $x$. I could also say Statement 3:

If $x=55$, then $4x+3$ is odd.

The power of a construction like Statement 1 is that it implies Statements 2 and 3: once we've proven it, we can use it for any integer we'd like. Statement 1 is not intended to be used for a specific value of $x$ that we decided not to name -- it can be used for any integer $x$ we want.

A side-note is that we can also apply such a statement to other "generic" integers. Consider Statement 4:

If $t$ is a positive integer, then $2^{t-1}$ is an integer.

We can combine the two to say that, if $t$ is a positive integer, then $2^{t-1}$ is an integer, and so if we set $x=2^{t-1}$ in Statement 1, we get that $4\cdot 2^{t-1}+3$ is an integer. So, Statement 4 and Statement 1 can be combined to give another general statement:

If $t$ is a positive integer, then $2^{t+1}+3$ is odd.

If we only cared about Statement 1 for a single integer $x$, doing something like the above wouldn't really be possible.

Carl Schildkraut
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  • Let me explain why my method does enable you to generate general proofs. Let the symbol $t$ represent one of the positive integers (although it is not clear which one). Then, $t - 1 \geq 0$ and $t - 1$ is an integer. So, $2^{t - 1}$ must be an integer. Now, since throughout the entire proof, the only information we used about the object represented by $t$ was that it is a positive integer, we can represent any positive integer using $t$, and the proof will hold. This last step is what enables the proof's generality. – Pranav Jain Dec 19 '20 at 03:27
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    @PranavJain Yes, something like that is right, although I'd avoid saying "although it is not clear which one." A statement like that would mean that, in fact, it is a particular integer, and we just don't know it yet, while in truth it can actually be any integer. This is the meaning of a "variable." – Carl Schildkraut Dec 19 '20 at 03:28
  • Would you classify the $t$ in this case as a variable? – Pranav Jain Dec 19 '20 at 03:32
  • @PranavJain Yes. If memory serves, this could be called an "independent variable." – Carl Schildkraut Dec 19 '20 at 03:33
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    Ok thanks. Just one final clarification. So, whenever I read something like "Let $x$ be an arbitrary integer" in a proof, can I in my head just think "Let $x$ be one of the integers"? Then, later I can consider generalizing on $x$ because the only info I used is that $x$ is an integer. I just want a quick way of thinking about this as I am studying comp. sci., which involves lots of proofs. – Pranav Jain Dec 19 '20 at 03:37
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    @PranavJain I think that makes sense. Regardless of exactly what sentence you think when you read this, it seems that you understand what's happening. – Carl Schildkraut Dec 19 '20 at 03:39