Definition:Symmetric Difference

Definition
The symmetric difference between two sets $$S$$ and $$T$$ is written $$S * T$$ and is defined as:
 * $$S * T = \left({S \setminus T}\right) \cup \left({T \setminus S}\right)$$

where:


 * $$S \setminus T$$ is the set difference between $$S$$ and $$T$$, defined as $$S \setminus T = \left\{{x: x \in S \and x \notin T}\right\}$$;
 * $$S \cup T$$ is the union of $$S$$ and $$T$$, defined as $$S \cup T = \left\{{x: x \in S \or x \in T}\right\}$$.

The symmetric difference can also be expressed as the set difference between their union and intersection:


 * $$S * T = \left({S \cup T}\right) \setminus \left({S \cap T}\right)$$

as is proved here.

Illustration by Venn Diagram
The red area in the following Venn diagram illustrates $$S * T$$:


 * VennDiagramSymmetricDifference.png

Alternative Names
The symmetric difference is also known as the disjount union, for intuitively obvious reasons.

Notation
There is no standard symbol for symmetric difference. The one used here: $$*$$ has been chosen somewhat arbitrarily; it's the one found by the author in the nearest work to hand.

The following are often found for $$S * T$$:


 * $$S \oplus T$$;
 * $$S + T$$;
 * $$S \triangle T$$.