Business Mathematics and Basic Statistics · Ch 9 — Sets — Operations and Functions

Set Operations — Union and Intersection

Just as numbers can be combined using addition and multiplication, sets can be combined using set operations. The two most basic ones are union and intersection.

The union of two sets $A$ and $B$ , written $A \cup B$ , is the set of all elements that belong to $A$ , or to $B$ , or to both:

A \cup B = \{x : x \in A \text{ or } x \in B\}.

The intersection of $A$ and $B$ , written $A \cap B$ , is the set of elements common to both $A$ and $B$ :

A \cap B = \{x : x \in A \text{ and } x \in B\}.

For example, if $A = \{2, 4, 6, 8, 10\}$ and $B = \{4, 8, 12, 16\}$ , then

A \cup B = \{2, 4, 6, 8, 10, 12, 16\}, \qquad A \cap B = \{4, 8\}.

Every element that appears in $A$ or $B$ (or both) is collected once into $A \cup B$ — note that a shared element such as $4$ is written only once in the union, since a set never lists the same element twice; only elements genuinely appearing in both lists make it into $A \cap B$ .

Note

A quick property to notice

$n(A \cup B)$ is generally not simply $n(A) + n(B)$ , because any elements shared between $A$ and $B$ would then be counted twice. In the example above, $n(A) = 5$ and $n(B) = 4$ , but $n(A \cup B) = 7$ , not $9$ — the two shared elements, $4$ and $8$ , must be subtracted once. This idea returns, fully developed, as the inclusion-exclusion principle later in this chapter. …

Definition 1Union of Two Sets

$A \cup B = \{x : x \in A \text{ or } x \in B\}$ — every element belonging to $A$ , to $B$ , or to both, …

Definition 2Intersection of Two Sets

$A \cap B = \{x : x \in A \text{ and } x \in B\}$ — every element common to bot …

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