Set Operations — Union, Intersection, Difference and Complement

Given sets $A$ and $B$ drawn from a universal set $U$, four operations combine or compare them.

Union ($A \cup B$): the set of elements that belong to $A$, or to $B$, or to both:

Intersection ($A \cap B$): the set of elements common to both $A$ and $B$:

If $A \cap B = \emptyset$, the two sets are called disjoint — they share no element at all.

Difference ($A - B$, also written $A \setminus B$): the set of elements in $A$ that are not in $B$:

Note that $A - B$ and $B - A$ are, in general, different sets.

Complement ($A'$, also written $A^c$ or $U - A$): everything in the universal set that is not in $A$:

For example, with $U = \{1,2,\dots,10\}$, $A = \{2,4,6,8,10\}$ and $B = \{3,6,9\}$:

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