Concept of Set

Basic concepts of set

1. The empty set. The set that does not contain any element is called an empty set, denoted by $\emptyset$ or $\{\}$.
2. Universal set. A set which contains all elements under consideration. It is denoted by $\mu$.
3. Equality. Two sets are equal if they have the same elements. i.e. If $A = \{a, b, c, d\}$ and $B = \{b, c, d, a\}$
4. Equivalent. Two sets are equivalent if they have the same number of elements. i.e. $A = \{a, b, c, d, e\}$ and $B = \{2, 1, 4, 5, 6\} \quad \therefore A \equiv B$
5. Subset. $A$ is a subset of $B$ if every member of $A$ is also a member of $B$. It is denoted by $A \subseteq B$
6. Proper subset. Suppose $A = \{a, b, c\}$ and $B = \{a, b, c, d\} \quad A \subset B$
7. Improper subset. A set is an improper subset of itself. $A \subseteq A$

Note i) $\emptyset \subseteq A$ (an empty set is a subset of any set) ii) $A \subseteq A$ (a set is a subset of itself)

Number of subsets in a set

Let $S = \{a, b, c\}$ How many subsets does it have?

The subsets are: $\{ \}, \{a, b, c\}, \{a\}, \{b\}, \{c\}, \{a, b\}, \{b, c\}, \{a, c\}$ $\rightarrow$ There are 8 subsets of S.

If $A = \{1\}$ and If $B = \{1,2\}$ Subset of $A$ are: $\{ \}$ and $\{1\}$ Subsets of $B$ are: $\{1\}, \{2\}, \{ \}, \{1,2\}$ Number of subsets of $A = 2$ Number of subsets of $B = 4$

If a set has $n$ members, the number of subsets $= 2^n$

The power set

Is a set which contains all subsets of the given set.

If $A = \{1\}$, subsets are $\{ \}, \{1\}$ Power set of $A$ is given by $P(A) = \{\{ \}, \{1\}\}$

Given $B = \{a, b, c\}$ The power set of $B$ is given by

$$P(B) = \{ \{a\}, \{b\}, \{c\}, \{a, b\}, \{b, c\}, \{a, c\}, \{a, b, c\}, \{ \} \}$$

Operation of sets

1. Union

The union of two sets A and B is denoted by $A \cup B$

• $A \cup B = \{ x: x \in A \text{ or } x \in B \}$
• Is a set which has elements of set A or set B without repetition.

Examples

→ If $A = \{a, b, c, d\}$ and $B = \{4, 7\}$ $A \cup B = \{a, b, c, d, 4, 7\}$

$$\text{If } A = \{a, b, c, d\} \text{ and } B = \{c, d, e, f\}$$ $$A \cup B = \{a, b, c, d, e, f\}$$

2. Intersection

Is a set which has both elements contained in set A and set B

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

Examples

$$\text{If } A = \{a, b, c, d\} \text{ and } B = \{a, f, g, d\}$$ $$A \cap B = \{a, d\}$$ $$\text{If } A = \{a, b, c\} \text{ and } B = \{1, 2, 3\}$$ $$A \cap B = \{\}$$

Here A and B are disjoint sets.

3. Complement

The complement of Set A denoted by $A'$ is the set of all elements which are in universal set but not in A.

E.g. $A = \{1, 2, 3, 4\}$

$$\mu = \{1, 2, 3, 4, 5, 6, 7\}$$

$A' = \{5, 6, 7\}$

4. Relative complement

Relative complement of A with respect to set B is denoted by $A \setminus B$ and is defined as follows:

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

Example

$$\begin{aligned} A &= \{a, b, c, d\} \\ B &= \{a, b, f, g, h\} \\ \text{Then } A \setminus B &= \{c, d\} \\ B \setminus A &= \{f, g, h\} \end{aligned}$$

5. The symmetric difference

All elements which are either in set A or set B but not both.

• The symmetric difference of A and B is denoted by $A \oplus B$

$$A \oplus B = \{x: x \in A \text{ or } x \in B \text{ but } x \notin A \cap B\}$$

Examples

$$A = \{a, b, c, d, e\}$$ $$\begin{aligned} B &= \{c, d, e, f, g\} \\ A \oplus B &= \{a, b, f, g\} \end{aligned}$$

Questions

1. List the subsets of the following sets

a) $A = \{2, 3, 4\}$

b) $B = \{a, \emptyset\}$

2. Let $A = \{x \in \mathbb{Z}^+ : 8 \leq x \leq 10\}$

Write down the subsets of A

3. Which of the following are true and which are false?

a) $\emptyset \subseteq \emptyset$

b) $0 = \emptyset$

c) $\emptyset \in \{0\}$

d) $\emptyset \subseteq \{0\}$

4. Let $A = \{1, 2, \{6, 7, 9\}, 10\}$

a) Is $\{\{2\}\} \in A$

b) Is $2 \in A$

c) Is $\{6, 7, 9\} \in A$

d) Is $\{1, \{6, 7, 9\}\} \subset A$

e) Is $\{2\} \in A$

f) Is $\{2\} \subset A$

5. Let $\mu$ be the set of all positive integers, A is the set of all even integers and B is the set of all odd integers. What are $A \cup B$ and $A \cap B$?