Types of Sets

CharliePark·2020년 9월 1일
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TIL

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Types of Sets

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

n()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}A = \{2, 4, 5\}, B = \{9, 11, 13\}

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

 

 

Subsets

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

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

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

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

A={1,2},B={1,2,3}then, ABA = \{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}then, P(A)={{},{1},{2},{3}.{1,2},{1,3},{2,3},{1,2,3}}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))=2kn(A) = k,\ then\ n(P(A)) = 2^k

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