Mathematics and Statistics · Ch 1 — Sets and Relations

Operations on Sets

Sets can be combined to form new sets. These operations are best pictured with a Venn diagram, in which the universal set $U$ is drawn as a rectangle and the sets inside it as circles; overlapping circles show elements shared between sets.

Union. The union $A \cup B$ is the set of elements in $A$ , or in $B$ , or in both:

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

Intersection. The intersection $A \cap B$ is the set of elements common to both:

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

If $A \cap B = \varnothing$ , the sets have no element in common and are called disjoint.

Figure 1 — Venn diagram of A = {1, 2, 3, 4} and B = {3, 4, 5, 6}: the overlap is the intersection A ∩ B = {3, 4}, and the whole shaded region is the union A ∪ B = {1, 2, 3, 4, 5, 6}

Difference. The difference $A - B$ (also written $A \setminus B$ ) is the set of elements in $A$ but not in $B$ :

A - B = \{x : x \in A \text{ and } x \notin B\}

In general $A - B \ne B - A$ .

Complement. Relative to a universal set $U$ , the complement $A'$ (or $A^{c}$ ) is the set of all elements of $U$ that are not in $A$ :

A' = U - A = \{x : x \in U \text{ and } x \notin A\}

Key algebraic laws of sets (all provable from the definitions, and easy to check on a Venn diagram):

Note

Laws of set operations

Commutative: $A \cup B = B \cup A$ , and $A \cap B = B \cap A$ .
Associative: $(A \cup B) \cup C = A \cup (B \cup C)$ ; similarly for $\cap$ .
Distributive: $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$ ; and $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$ .
De Morgan's laws: $(A \cup B)' = A' \cap B'$ , and $(A \cap B)' = A' \cup B'$ . …

Definition 1Union $A \cup B$

The set of all elements belonging to $A$ or $B$ ( …

Definition 2Intersection $A \cap B$

The set of elements common to both $A$ and $B$ . If empty, the sets a …

Definition 3Difference $A - B$

The set of elements in $A$ but not in $B$ ; in general $A - B …

Definition 4Complement $A'$

The set $U - A$ of all elements of the universal set $U$ that are …

Definition 5Inclusion–exclusion

$n(A \cup B) = n(A) + n(B) - n(A \cap B)$ , correcting for double-counting …

Start your 14-day free trial to unlock the full solution →

← Previous · 2Types of Sets Next · 4 →Ordered Pairs and Cartesian Product