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What is the use of $ \Delta $ in set theory?

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  • $\begingroup$ Besides symmetric difference (the answers showing so far), there is the $\Delta$-lemma used in forcing arguments and the diagonal intersection of a transfinite sequence of sets of ordinal numbers. $\endgroup$ Jun 2, 2014 at 15:56

2 Answers 2

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The $\Delta$ in set theory is the symmetric difference of two sets.

$A$ $\Delta$ $B$ $=$ $(B-A) \cup (A-B)$

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    $\begingroup$ And the symbol that should be better used is $\triangle$. $\endgroup$
    – Pedro
    Nov 14, 2013 at 14:01
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In the context of elementary set theory the symbol $\triangle$ usually denotes the operation of symmetric difference of two sets: if $A$ and $B$ are sets, $$A\mathrel{\triangle}B=(A\setminus B)\cup(B\setminus A)\;.$$

This definition explains the name symmetric difference: we take both the set difference $A\setminus B$ and the set difference $B\setminus A$ and then form their union, so that the operation is commutative:

$$A\mathrel{\triangle}B=B\mathrel{\triangle}A\;.$$

You ought to prove to yourself that

$$A\mathrel{\triangle}B=(A\cup B)\setminus(A\cap B)\;;$$

this is even sometimes used as the definition of $A\mathrel{\triangle}B$.

In plain English, $A\mathrel{\triangle}B$ is the set of things that belong to exactly one of $A$ and $B$.

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  • $\begingroup$ Might be worthwhile to mention associativity, and point to the fact that $(\mathcal P(X), \Delta)$ is an abelian group with unit $\varnothing$. $\endgroup$
    – Lord_Farin
    Nov 14, 2013 at 13:43
  • $\begingroup$ @Lord_Farin: I’d rather not: at this point that seems more likely to get in the way than to help. Thanks for catching the dropped letter. $\endgroup$ Nov 14, 2013 at 13:45
  • $\begingroup$ @Lord_Farin Ring with $\cap$! $\endgroup$
    – Pedro
    Nov 14, 2013 at 14:01

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