( 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