Set Operations (Difference & Complement)
A−B={x ∣ x∈A and x∈B} : Difference
Ac=U−A : Complement (A‘)
Properties of Difference and Complement
A∪Ac=U
(Ac)c=A
Uc={ },{ }c=U
A−B=A∩Bc
De Morgan's Law
(A∪B)c=Ac∩Bc
(A∩B)c=Ac∪Bc
Partition of Sets
Disjoint Sets : Two sets are called disjoint if they have no elements in common A∩B=∅
Mutually Disjoint Sets : A1,A2,A3,A4,⋯ 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,⋯} is a partition of a set A if and only if two conditions are met.
Two conditions are :
- A is the Union of {A1,A2,A3,⋯}
- {A1,A2,A3,⋯} are mutually disjoint
e.g. A={1,2,3,4,5,6},then {{1,2},{3},{4,5,6}} is a partition of A