
# 2.2: Operations with Sets

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

#### Complements of Sets

Definition

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

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

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

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

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

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

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

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

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

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

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

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

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 $$B = \{1, 3, 5, 7\}$$

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