Difference & Complement, Partition

Set Operations (Difference & Complement)

$A-B = \{x\ |\ x \in A\ and\ x \not \in B\}$ : Difference

$A^c = U - A$ : Complement $(A`)$

 

Properties of Difference and Complement

$A \cup A^c = U$

$(A^c)^c = A$

$U^c = \{\ \}, \{\ \}^c = U$

$A-B = A \cap B^c$

 

De Morgan's Law

$(A \cup B)^c = A^c \cap B^c$

$(A \cap B)^c = A^c \cup B^c$

 

Partition of Sets

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

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

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

Two conditions are :

1. $A$ is the Union of $\{A_1, A_2, A_3, \cdots \}$
2. $\{A_1, A_2, A_3, \cdots \}$ are mutually disjoint

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