FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS I 2.7
- The union of any set A and its complement A ́ is the universal set, i.e., A ∪ A ́ = U.
- The intersection of any set and its complement A ́ is the null set, i.e. , A ∩ A ́ = φ.
As for example : U = {1, 2, 3, ........., 10}, A = { 2, 4, 7}
A ́ ( = U ~ A) = {1, 3, 5, 6, 8, 9, 10} = U, A ∩ A ́ = φ
Again (A ́) ́ = {2, 4, 7) = A, (i.e., complement of the complement of A is equal to A itself.
U ́ = φ , (i.e., complement of a universal set is empty).
Again the complement of an empty set is a universal set, i.e., φ ́ = U.
If A ⊂ B then B ́ ⊂ A ́ for set A and B.
Complement of A is represented by shaded region.
Symmetric Difference :
For the two sets A and B, the symmetric difference is (A ~ B) ∪ ( B ~ A)
and is denoted by A ∆ B (read as A symmetric difference B)
As for example : Let A = { 1, 2, 3, 4,8}, B = {2, 4, 6, 7}.
Now, A ~ B = { 1, 3, 8}, B ~ A = {6, 7}
∴A ∆ B = { 1, 3, 8} ∪ (6,7) = {1, 3, 6, 7, 8}
By Venn diagram :
A ∆ B is represented by shaded region. It is clear that A ∆ B denotes the set of all those elements that belong
to A and B except those which do not belong to A and B both, i.e., is the set of elements which belongs to
A or B but not to both.
A B∩
A~B B~A
A B∩
Difference between :
φ, (0) and {φ}
φ is a null set.
{0} is a singleton set whose only element is zero.
{φ} is also a singleton set whose only element is a null set.
2.1.3 Properties :
- The empty set is a sub-set of any arbitrary set A.
- The empty set is unique.
Note :