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What does it mean when you have a set like A = {1, {2}, 3, 4}? Is {2} a subset within A?

Say I have the following:

A = {1, {2}, 3, 4}, B = {1, 2, {3}}, C = {1, 2}

How many elements are in $A \cap B \cap C$? 1 or 2?

What if I want to find $A \times B$, is there 6 elements or 12?

Asaf Karagila
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gator
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2 Answers2

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The best way to deal with this is to write $a=\{2\}$ and $b=\{3\}$. Now we have: $$A=\{1,a,3,4\}, B=\{1,2,b\}, C=\{1,2\}.$$

In the common assumptions, it's not the case that $X=\{X\}$ or even $X\in X$. So it's safe to assume that $2\neq\{2\}$, and so on. Do note, however, that in some mathematical contexts it might be the case, but when you get to that, you usually know already what is the context and assumptions.

Asaf Karagila
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The set $A$ has $4$ elements: $1$, $\{2\}$, $3$, and $4$. The set $B$ has $3$ elements: $1$, $2$, and $\{3\}$. Thus, $A\times B$ has $4\cdot3=12$ elements. As you can see from those lists, the only element of $A\cap B$ is $1$; $1$ is also an element of $C$, so $A\cap B\cap C=\{1\}$. (For the record, $B\cap C=\{1,2\}$: $C\subseteq B$.)

Brian M. Scott
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  • But what does element 2 of A mean? Why is {2} and {3} not counted when you find the intersect of A and B? Is {2} different from 2? – gator Nov 01 '13 at 10:46
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    @riista: $2$ is not an element of $A$: ${2}$ is an element of $A$. ${2}$ is the set whose only member is $2$; this is not the same thing as the number $2$. The set ${2}$ is an element of $A$ but not of $B$, and the number $2$ is an element of $B$ but not of $A$. – Brian M. Scott Nov 01 '13 at 10:49
  • @riista Maybe this http://math.stackexchange.com/questions/527819/element-of-subset-of-and-empty-sets/528184#528184 is helpful. – Michael Hoppe Nov 01 '13 at 13:20
  • Shouldn't that say $A \cap B = {1}$ in the last sentence? Maybe that is what the OP's comment was about. – Trevor Wilson Nov 01 '13 at 15:49
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    @Trevor: No, it was supposed to say that $B\cap C={1,2}$. Fixed; thanks. – Brian M. Scott Nov 01 '13 at 15:55