Types of Sets and Subsets

The Empty Set

A set which does not contain any element is called the empty set (or the null set or the void set). It is denoted by the symbol $\phi$ or $\{\}$.

• Example: $A = \{x : 1 < x < 2, x \text{ is a natural number}\}$. Since there is no natural number between 1 and 2, A is the empty set, so $A = \phi$.

Finite and Infinite Sets

• A set which is empty or consists of a definite number of elements is called a finite set.
• Otherwise, the set is called an infinite set.
• Example (Finite): The set of days in a week, W, is finite because it has 7 elements.
• Example (Infinite): The set of points on a line, G, is infinite because there are infinitely many points on a line.

Equal Sets

Two sets A and B are said to be equal if they have exactly the same elements. We write this as $A = B$.

• Example: If $A = \{1, 2, 3, 4\}$ and $B = \{3, 1, 4, 2\}$, then $A = B$ because both sets contain the exact same elements.
• Repetition of elements does not change a set. For example, $\{1, 2, 3\}$ is equal to $\{2, 2, 1, 3, 3\}$.

Subsets

A set A is said to be a subset of a set B if every element of A is also an element of B. This is denoted by $A \subset B$.

• In other words: $A \subset B$ if $a \in A \implies a \in B$.
• Important Rules:
  • Every set is a subset of itself ($A \subset A$).
  • The empty set $\phi$ is a subset of every set ($\phi \subset A$).
• Proper Subset: If $A \subset B$ and $A \ne B$, then A is called a proper subset of B.
• Singleton Set: A set with only one element is called a singleton set, e.g., $\{a\}$.

Subsets of Real Numbers (Intervals): Intervals are a common way to represent subsets of the set of real numbers ($\mathbb{R}$).

• Open Interval: $(a, b) = \{y : a < y < b\}$
• Closed Interval: $[a, b] = \{x : a \le x \le b\}$
• Semi-Open/Closed:
• $[a, b) = \{x : a \le x < b\}$
• $(a, b] = \{x : a < x \le b\}$

Example 1: Identifying Set Types

Question: State which of the following sets are finite or infinite:

(i) $\{x : x \in \mathbb{N} \text{ and } (x-1)(x-2) = 0\}$

(ii) $\{x : x \in \mathbb{N} \text{ and x is prime}\}$

Solution: (i) The solutions to $(x-1)(x-2)=0$ are $x=1$ and $x=2$. The set is $\{1, 2\}$. This set has a definite number of elements (2), so it is finite.

(ii) The set of prime numbers is $\{2, 3, 5, 7, 11, ...\}$. This list continues indefinitely. Therefore, the set is infinite.

Example 2: Checking for Subsets

Question: Consider the sets $A=\{1,3\}$, $B=\{1,5,9\}$, $C=\{1,3,5,7,9\}$. Insert the symbol $\subset$ or $\not\subset$ between the following pairs:

(i) A … B

(ii) A … C

(iii) B … C

Solution: (i) $A \not\subset B$, because $3 \in A$ but $3 \notin B$.

(ii) $A \subset C$, because all elements of A (1 and 3) are also present in C.

(iii) $B \subset C$, because all elements of B (1, 5, and 9) are also present in C.