Types of sets

An empty set is a set with no elements. It is denoted by the symbol $\emptyset$ or in roster form as { }. Example:

• $A = \{ \text{three-sided rectangles} \} \Rightarrow A = \emptyset$
• $B = \{ x : 7 < x < 8, x \in \mathbb{N} \} = \emptyset$ (no natural number exists between 7 and 8)

Example 1

Given: $K = \{ x : -10 \leq x \leq -1 \}$, find $x \in \mathbb{N}$. Solution: No natural numbers exist between -10 and -1. So, $K = \emptyset$.

Example 2

Given $P = \{1, 3, 5, 7, 9, 11\}$, identify even numbers in $P$. Solution: None of the elements are even. So, $E = \emptyset$.

Example 3

Given $A = \{\text{Hemedi, Shedrack, Peter, Jonathan, Patrick}\}$, find the set of girl names $G \subset A$. Solution: No girl names present. Hence, $G = \emptyset$.

Two sets are equal if they contain exactly the same elements. For example: $A = \{1,2,3,4,5\}, B = \{1,2,3,4,5\} \Rightarrow A = B$

Example 1

Given $A = \{\text{Halima, Angelina, Christian, Sarapia}\}, B = \{\text{Halima, Angelina, Christian, Sarapia}\}$ Since all elements match, $A = B$.

Unequal sets have at least one differing element. Denoted as $C \ne D$. Example: $C = \{a, b, c, d, e, g\}, D = \{a, b, c, d, e, f\} \Rightarrow C \ne D$

Example 2

$C = \{2, 4, 6, 8\}, D = \{2, 4, 6, 8, 10, 12\} \Rightarrow C \ne D$

Two sets are equivalent if they have the same number of elements. Denoted as $A \equiv B$. Example: $A = \{a, e, i, o, u\}, B = \{1, 2, 3, 4, 5\} \Rightarrow n(A) = n(B) = 5 \Rightarrow A \equiv B$

Example 1

$S = \{1,2,3,4,5,6\}, G = \{x \in \mathbb{Z} : 1 \leq x \leq 6\} \Rightarrow S \equiv G$

Example 2

$D = \{1, 3, 5, 7, 9, 11, 13\}, G = \{x \in \mathbb{N} : 3 \leq x \leq 9\} \Rightarrow D \not\equiv G$ since $n(D) \ne n(G)$

A set $A$ is a subset of set $B$ if all elements of $A$ are also in $B$. Denoted $A \subset B$. The number of subsets of a set with $n$ elements is $2^n$.

Proper vs improper subsets

• Proper subset: Contains some but not all elements of a set, $A \subset B$
• Improper subset: Contains all elements of the set itself, $A \subseteq A$

The power set of a set is the set of all its subsets, denoted $P(A)$. If $n(A) = n$, then $|P(A)| = 2^n$.

Example 1

Given $B = \{1, 2, 3\}$

• All subsets: $\{\}, \{1\}, \{2\}, \{3\}, \{1,2\}, \{1,3\}, \{2,3\}, \{1,2,3\}$
• Improper subset: $\{1,2,3\}$
• Proper subsets: All others
• $P(B) = \{\emptyset, \{1\}, \{2\}, \{3\}, \{1,2\}, \{1,3\}, \{2,3\}, \{1,2,3\}\}$
• $n(P(B)) = 8$

Example 2

$K = \{\text{Anna, Ally, Halima, John}\}$. Total subsets: $2^4 = 16$

A finite set has countable elements. An infinite set has uncountable or unlimited elements.

• Example of finite set: $V = \{a, e, i, o, u\}$
• Example of infinite set: $P = \{\text{prime numbers}\}$

Example 1

$A = \{y : y = 2^n, y \in \mathbb{N}\}$, $B = \{1,2,3,4,5,6,\ldots\}$, $C = \{x : 1 \leq x \leq 20, x \in \mathbb{Z}\}$ → A and B are infinite, C is finite

A singleton set contains only one element.

• $A = \{0\}$
• $B = \{k\}$
• $C = \{x : x = 3 \text{ and } x \in \mathbb{N} \} = \{3\}$

A universal set contains all elements under consideration, denoted by $U$ or $\xi$. Example: If $A = \{1,2,a,4,5\}, B = \{1,2,3,a,b,c,d\}$, then $U = \{1,2,3,4,5,a,b,c,d\}$ $A \subset U, B \subset U$

Example 1

Given: $U = \{7, 8, 9, 10, 11, 12, 13, 14, 15, 16\}$

• $A = \{x \in U : x \text{ is a factor of 60} \} = \{10, 12, 15\}$
• $B = \{x \in U : x \text{ is even} \} = \{8, 10, 12, 14, 16\}$
• $C = \{x \in U : x \text{ is odd} \} = \{7, 9, 11, 13, 15\}$