Difference & Complement, Partition

CharliePark·2020년 9월 4일
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Set Operations (Difference & Complement)

AB={x  xA and x∉B}A-B = \{x\ |\ x \in A\ and\ x \not \in B\} : Difference

Ac=UAA^c = U - A : Complement (A)(A`)

 

Properties of Difference and Complement

AAc=UA \cup A^c = U

(Ac)c=A(A^c)^c = A

Uc={ },{ }c=UU^c = \{\ \}, \{\ \}^c = U

AB=ABcA-B = A \cap B^c

 

De Morgan's Law

(AB)c=AcBc(A \cup B)^c = A^c \cap B^c

(AB)c=AcBc(A \cap B)^c = A^c \cup B^c

 

Partition of Sets

Disjoint Sets : Two sets are called disjoint if they have no elements in common AB=A \cap B = \varnothing

Mutually Disjoint Sets : A1,A2,A3,A4,A_1, A_2, A_3, A_4, \cdots if, and only if, no two sets have an element in common

Partition of Sets : A finite or infinite collection of non-empty sets {A1,A2,A3,}\{A_1, A_2, A_3, \cdots \} is a partition of a set AA if and only if two conditions are met.

Two conditions are :

  1. AA is the Union of {A1,A2,A3,}\{A_1, A_2, A_3, \cdots \}
  2. {A1,A2,A3,}\{A_1, A_2, A_3, \cdots \} are mutually disjoint

e.g. A={1,2,3,4,5,6},then {{1,2},{3},{4,5,6}} is a partition of Ae.g.\ A = \{1, 2, 3, 4, 5, 6\},\\ then\ \{\{1, 2\}, \{3\}, \{4, 5, 6\}\}\ is\ a\ partition\ of\ A

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