2.2: Set Operations

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May 20, 2022
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- 2.1: Subsets and Equality
- 2.3: Venn Diagrams and Euler Diagrams

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Complement of Sets

Definition: Complement

The complement of a set is another set which contains only elements not found in the first set.

Let $A$ be a set.

$A$ ^c = $\{x \mid x \notin A\}$

We write ^c to denote a complementary set.

Often, the context provides a "universe" of all possible elements pertinent to a given discussion. Suppose we have given such a set of "all" elements. Let us call it $U$ . Then, the complement of a set $A$ , denoted by $A^c$ , is deﬁned as $A^c = U - A$ . In our work with sets, the existence of a universal set $U$ is tacitly assumed.

Example $\PageIndex{1}$

Consider $\mathbb{Q}$ and $\mathbb{Q}$ ^c, the sets of rational and irrational numbers, respectively:

$x \in \mathbb{Q} \to x \notin \mathbb{Q}$ ^c, since a number cannot be both rational and irrational.

So, the sets of rational and irrational numbers are complements of each other.

Union

Definition: Union

A union of two sets creates a "united" set containing all terms from both sets.

$A \cup B = \{x \mid (x \in A) \vee (x \in B)\}$

$alt$

Example $\PageIndex{2}$

Let $A = \{1, 3, 5\}$ and $B = \{2, 4, 6\}$

Then $A \cup B = \{1, 2, 3, 4, 5, 6 \}$

Intersection

Definition: Intersection

The intersection of two sets creates a set with elements that are in both sets.

$A \cap B = \{x \mid (x \in A) \wedge (x \in B)\}$

$alt$

Example $\PageIndex{3}$

Let $A = \{8, 12, \frac{3}{7}, -22\}$ and $B = \{8 675 309, 42, 12, 8, 57\}$

Then $A \cap B = \{8, 12\}$

Set Difference

Definition: Set difference

The difference between two sets generates a set which has no elements of the second set.

$A - B = \{x \mid (x \in A) \wedge (x \notin B)\}$

$alt$

Example $\PageIndex{4}$

Let $A = \{8, 12, \frac{3}{7}, -22\}$ and $B = \{8 675 309, 42, 12, 8, 57\}$ .

Then $A - B = \{\frac{3}{7}, -22\}$

The Empty Set

Definition: Empty set

The empty set is a set that has no elements. It is written $\{\}$ or $\emptyset$ .

$\emptyset \subseteq A$ , for any set A

The empty set has just one subset, which is itself. The empty set is also a subset of every set, since a set with no elements naturally fits into any set with elements.

Disjoints

Definition: Disjoint sets

A and B are called disjoints if $A \cap B = \emptyset$ .

Example $\PageIndex{5}$

Consider sets $\mathbb{Q}$ and $\mathbb{Q}$ ^c:

Since $\mathbb{Q} \cap \mathbb{Q}$ ^c $= \emptyset$ , these sets are called disjoints.

Cartesian Product

Definition: Cartesian products

The so-called Cartesian product of sets is a powerful and ubiquitous method to construct new sets out of old ones.

Let $A$ and $B$ be sets. Then the Cartesian product of $A$ and $B$ , denoted by $A \times B$ , is the set of all ordered pairs $(a, b),$ with $a \in A$ and $b \in B.$ In other words,

$A \times B = \{(a, b) ~|~ a \in A, b \in B\} .$

An important example of this construction is the Euclidean plane $\mathbb{R}^2 = \mathbb{R} \times \mathbb{R}$ . It is not an accident that $x$ and $y$ in the pair $(x, y)$ are called the Cartesian coordinates of the point $(x, y)$ in the plane.

Example $\PageIndex{6}$

Let $A = \{2, 4, 6, 8\}$ and $B = \{1, 3, 5, 7\}$ . then

$A \times B = \{(2, 1), (4, 3), (6, 5), (8, 7)\}$