( iii) φ ∩ A = φ, U ∩ A = A( Law of φ and U).( iv) A ∩ A = A( Idempotent law)
( v) A ∩( B ∪ C) =( A ∩ B) ∪( A ∩ C)( Distributive law) i. e., ∩ distributes over ∪
Difference of Sets
The difference of the sets A and B in this order is the set of elements which belong to A but not to B. Symbolically, we write A – B and read as“ A minus B”. A – B = { x: x ∈ A and x ∉ B }
Example Let A = { 1, 2, 3, 4, 5, 6 }, B = { 2, 4, 6, 8 }. Find A – B and B – A Solution: A-B = { 1,3,5 } B-A ={ 8 }
Note that The sets A – B, A ∩ B and B – A are mutually disjoint sets.
Complement of a Set
Let U be the universal set and A a subset of U. Then the complement of A is the set of all elements of U which are not the elements of A. Symbolically, we write A ′ to denote the complement of A with respect to U. Thus, A ′ = { x: x ∈ U and x ∉ A }. Obviously A ′ = U – A