# What is the use of Delta symbol in set theory?

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

• 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. Jun 2, 2014 at 15:56

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

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

• And the symbol that should be better used is $\triangle$.
– Pedro
Nov 14, 2013 at 14:01

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 $$B$$.

• Might be worthwhile to mention associativity, and point to the fact that $(\mathcal P(X), \Delta)$ is an abelian group with unit $\varnothing$. Nov 14, 2013 at 13:43
• @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. Nov 14, 2013 at 13:45
• @Lord_Farin Ring with $\cap$!
– Pedro
Nov 14, 2013 at 14:01