Set Operations

Union

The set consisting of all elements that belongs to $A$ or $B$ or both.

Notation

$A\cup B=\{x\mid x\in A$ or $x\in B\}$

Example

$A=\{a,b,c,d,e,f\}B=\{b,d,r,s\}$

$\therefore A\cup B=\{a,b,c,d,e,f,r,s\}$

Intersection

Consists of all elements that belong to both $A$ and $B$, denoted by $A\cap B$.

Notation

$A\cap B=\{x\mid x\in A$ and $x\in B\}$

Example

$A=\{a,b,c,d,e,f\}B=\{b,e,f,r,s\}C=\{a,t,u,v\}$

$\therefore A\cap B=\{b,e,f\}A\cap C=\{a\}B\cap C=\varnothing$

$B$ and $C$ are called disjoint sets, which are sets that have no elements in common.

Complement

Set of all elements that belong to $A$ but not to $B$, denoted by $A-B$ (or $A\setminus B$).

Notation

$A-B=\{x|x\in A$ and $x\notin B\}$

Example

$A=\{a,b,c\}B=\{b,c,d,e\}$

$\therefore A-B=\{a\}B-A=\{d,e\}$

If $U$ is a universal set containing $A$, then $U\setminus A$ (or $\bar{A}$) is called the complement of $A$.

Notation

$\bar{A}=\{x|x\notin A\}$

Example

$U=\mathbb{Z}$

$A=\{x\mid x\text{ is an integer and }x\le 4\}$

$\therefore \bar{A}=\{x\mid x\text{ is an integer and }x>4\}$

Symmetric Difference

Set of all elements that belong to $A$ or to $B$, but not to both $A$ and $B$, denoted by $A\oplus B$.

Notation

$A\oplus B\{x\mid x(x\in A$ and $x\notin B)$ or $(x\in B$ and $x\notin A)\}$

Example

$A=\{a,b,c,d\}B=\{a,c,e,f,g\}$

$\therefore A\oplus B=\{b,d,e,f,g\}$