What dose the oplus operator do in set theory

Question

I'm having some trouble understand oplus in set theory, I couldn't find a simple explanation or example. So lets say you have $A=\{a,b,c,d\}$ and $B=\{a,c,e,g,i\}$

What would be $A \oplus B$ ? What does the operator do simply?

Answer 1

It is the symmetric difference operator, generally denoted by $\triangle$ but $\oplus$ is also a symbol for symmetric difference operator (Wikipedia_reference). For the meaning of the operator, we have $$A \oplus B = (A \cup B) \text{\\}(A \cap B) $$ Then what would be $A \oplus B$ in your example?

Answer 2

This symbol is also used in logic to denote the "exclusive or " ( or " XOR" ) operator.

The " + " is for the " OR " part of "X-OR" and the circle is for the " NOT-AND " part.

Indeed : "A XOR B" is equivalent to " (A OR B) AND ( NOT ( A&B) ) "

In the same way as the set algebra operation of union ( sumbol : U ) is defined using the " OR" operator, the symmetric difference operation ( symbol : circled + , or delta) is defined using the X-OR operator.

A U B = { x| x belongs to A OR x belongs to B }

A O-PLUS B

= A Delta B

= { x | (x belongs to A) XOR (x belongs to B) }

= { x | (x belongs to A OR B) & (x does not belong both to A & to B)}

Remark : I know this symbol " O plus" is still in use in engeneering ( at least in France) ; maybe it is a bit old fashioned in logic and set theory ( but I may be wrong).