Paper 4: Fundamentals of Business Mathematics & Statistic

FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS I 2.7

1. The union of any set A and its complement A ́ is the universal set, i.e., A ∪ A ́ = U.


2. 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 :

1. The empty set is a sub-set of any arbitrary set A.


2. The empty set is unique.
   Note :