Set Properties

General

Sets can have unrelated elements

$S=\{r,17,\text{Raphael},\text{Shenzhen}\}$

Repeated elements in the set are ignored

$A=\{1,3,2,3,1\}$

is this same as

$A=\{1,2,3\}$

The order of the elements is unaccounted for

$A=\{1,3,2\}$, $A=\{3,2,1\}$,

$A=\{3,1,2\}$, $A=\{2,1,3\}$

all have the same representations

Sets may have other sets as members

$A=\{\varnothing ,\{a\},\{b\},\{a,b\}\}$

Membership

If $x$ is an element of the set $A$, we use the symbol $\in$ to denote membership.

Example

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

$1\in A\text{ and }3\in A\text{, but }2\notin A$

Question

What are the members of set $B$?

$B=\{x\mid x\in R\text{ and }{x}^2=-1\}$

Answer

$\varnothing$, as there is no real number squared that would equal $-1$. They would have to be a complex number.

Equality

Two sets $A$ and $B$ are equal if they have the same elements, that is if and only if for all $x(x\in A\text{ if and only if x }\in B)$. Thus, we write $A=B$.

Example

$A=\{1,2,3\}$

$B=\{x\mid x\in Z^{+}\text{ and }{x}^2<12\}$

$\therefore A=B$

Cardinality

A set is called finite if it has $n$ distinct elements, where $n\in N$. $n$ is called the cardinality of $A$, denoted by $|A|$. An infinite set would have a cardinality of $n(A)=\infty$.

Example

$A=\{1,2,3,4,5\}\therefore |A|=5$

$S=\{x\mid x\text{ is a letter in the English alphabet }\}\therefore |S|=26$

$|\varnothing |=0$

Cartesian Product

The cartesian product of the sets $A_1,A_2,\dots ,A_n$ is the set of n-tuples $(a_1,a_2,\dots ,a_n)$ where $a_i$ belongs to $A_i$ for $i=2,2,\dots ,n$. It is denoted by the form: $A_1\times A_2\times \dots A_n$.

Example

$A=\{1,2\}B=\{a,b,c\}$

$A\times B=\{(1,a),(1,b),(1,c),(2,a),(2,b),(2,c)\}$

$B\times A=\{(a,1),(a,2),(b,1),(b,2),(c,1),(c,2)\}$

$A\times A=\{(1,1),(1,2),(2,1),(2,2)\}$

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

Question

$A=\{0,1\}B=\{1,2\}C=\{0,1,2\}$

Find $C\times B\times A$.

Answer

$$\begin{gathered} C\times B\times A=\{ \\ (0,1,0),(0,1,1),(0,1,2), \\ (0,2,0),(0,2,1),(0,2,2), \\ (1,1,0),(1,1,1),(1,1,2), \\ (1,2,0),(1,2,1),(1,2,2)\} \end{gathered}$$