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Question: What is A $\times$ A , where A = {0, $\pm$1, $\pm$2, ...} ?

Thinking: Is this a set say B = {0, 1, 2, ... } ?

This was in my homework can you help me ?

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3 Answers

up vote 4 down vote accepted

Your set $A$ is the set of integers $\mathbb{Z}$. So the product is $\mathbb{Z}\times \mathbb{Z}=\{(a,b);~a,b\in \mathbb{Z}\}$, the set of all ordered pairs of integer numbers.


ps: by the way, both sets have the same number of elements, that is, they have the same cardinality.

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If you consider $\mathbb{Z}\times \mathbb{Z}$ as a subset of $\mathbb{R}\times \mathbb{R}$, its complement is the set of ordered pairs with both coordinates real but non integer numbers. –  Sigur Oct 16 '12 at 22:47
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No, $A\times A$ is not a sequence. Neither is $A$: it’s just a set.

By definition $A\times A=\{\langle a_1,a_2\rangle:a_1,a_2\in A\}$. Thus, $A\times A$ contains elements like $\langle 1,0\rangle$, $\langle -2,17\rangle$, and so on. Indeed, since $A$ is just the set of all integers, usually denoted by $\Bbb Z$, $A\times A$ is simply the set of all ordered pairs of integers. Here is a picture of $A\times A$: the lines are the coordinate axes, and the dots are the points of $A\times A$, thought of as points in the plane.

enter image description here

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@DreamLighter, are you talking about the complement? Complement with respect to what? –  Sigur Oct 16 '12 at 22:45
    
@Sigur with respect to $R^2$ –  DreamLighter Oct 16 '12 at 22:47
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@DreamLighter: $\Bbb R^2\setminus(A\times A)$ is everything in the picture except the dots. It’s the plane with a regular grid of holes punched in it. But that really hasn’t anything to do with the original question. –  Brian M. Scott Oct 16 '12 at 22:48
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If this is your set theory homework you might want to know why it is a set. Well where did this notation $A \times A$ come from? Assuming you are working with ZFC, the cross product doesn't seem to be in the list of axioms. Instead, the cross product is defined using these axioms to mean precisely the set $\{(a,b) \mid a,b \in A\}$. So... by definition of the cross product we have $A \times A$ is a set (assuming $A$ itself is a set). Don't go down the path of "It's a set because I can write it as $\{a,b,c,...\}$". Doing so might lead you to think some things are sets which aren't!

Similarly you might ask what $(a,b)$ means... and that too is defined in terms of the ZFC axioms. Make sure you ask these questions (and answer them!) earlier rather than later.

Oh, and note that your original collection $A$ is in fact a set. If you guys didn't establish this in class then you might consider trying it for yourself. On the other hand, your teacher may want you to assume it is a set (for now, until you later show it to be a set rigorously).

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Great. But based on the level of the question I guess that ZF axioms is not the case here. +1 for your answer. –  Sigur Oct 16 '12 at 23:16
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@Sigur Thanks! I actually looked at the OPs profile and it seems like he/she might be in some set theoretic class... the question is worded pretty naively though I agree –  roliu Oct 17 '12 at 0:29
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