Types of Sets

Types of Sets

$\mathbb{U}$ : Universal set : set that contains everything

$\varnothing$ : Empty set (Null set) : set that contains nothing

Singleton Set : set that has one element

Finite Set : set that has finite number of elements

Infinite Set : set that has infinite amount of elements

Subset : sub of a set.

 

Cardinal number of a set

$A = \{1, 2, 3, 4, 5\},\quad n(A) = 5$

$n(\varnothing)$ = 0

 

Equivalent Set

if the cardinal number of two sets are same, they are equivalent sets

$A = \{2, 4, 5\}, B = \{9, 11, 13\}$

then A and B is equivalent. notation is $A \sim B$

 

 

Subsets

If $A$ and $B$ are sets, then $A$ is called a subset of $B$, written $A\subset B$, if, and only if, every element of $A$ is also an element of $B$

$A = \{1, 2\}, B = \{1, 2, 3\}\quad then,\ A \subseteq B$

$A \subseteq A$, every set is a subset of itself

$\varnothing \subseteq A$, empty set is a subset of every set

$A = \{1, 2\}, B = \{1, 2, 3\}\quad then,\ A \subset B$

A is subset of B and A is not equal to B, so A is proper subset of B

 

Power Sets

Power Set of A, denoted P(A), is the set of all subsets of A

$A = \{1, 2, 3\}\quad then,\ P(A) = \{\{\}, \{1\}, \{2\}, \{3\}. \{1,2\}, \{1, 3\}, \{2, 3\}, \{1, 2, 3\}\}$

so if $n(A) = k,\ then\ n(P(A)) = 2^k$