Set Operations

The universal set, denoted by $U$, is the set that contains all elements under consideration (that's not the set of everything, though.) Every set is a subset of $U$: $A \subseteq U$. Examples:

• If $U = \{1,2,3,4,5,6\}$ and $A = \{1,2,3\}$, then $A \subseteq U$.
• If $U = \mathbb{R}$ (the set of real numbers), then $\mathbb{N}$ (natural numbers) is a subset of $U$.
• If $U = \{a, b, c, d, e\}$ and $B = \{a, d, e\}$, then $B \subseteq U$.

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The complement of a set $A$, denoted by $A^c$ (

$A^c$

) or $\overline{A}$ (

$\overline{A}$

), consists of all elements in the universal set $U$ that are not in $A$: $A^c = \{ x \mid x \in U \,$$\text{ and } \,$$x \notin A \}$. Examples:

• If $U = \{1,2,3,4,5\}$ and $A = \{1,3,5\}$, then $A^c = \{2,4\}$.
• If $U = \mathbb{Z}$ and $A$ is the set of even integers, then $A^c$ is the set of odd integers.
• If $U = \{a, b, c, d\}$ and $A = \{a, c\}$, then $A^c = \{b, d\}$.