Operation on Sets:
Sets are pictorially represented by means of diagrams known as the Venn diagrams. In Venn diagrams, universal sets are denoted by rectangular boxes and their subsets by closed circles or ellipses inside the box.
Union- The union of two sets A and B is the set consisting of all the elements which are either in A or in B or in both and is denoted by A ∪ B. Thus if A = { 1, 2, 3 } and B = { 2, 4, 6 }, then A ∪ B = { 1, 2, 3, 4, 6 }.
Obviously, A ∪ B = { x : x ∈ A or x ∈ B}.
The Venn diagram for the union of two sets A and B are shown in (Fig. i).
If set B is a subset of A, then A ∪ B = A (Fig. ii). On the other hand, if A is a subset of B, then A ∪ B = B (Fig. iii).
In the case of a finite family of sets A1, A2, A3, …….An, their union is given by-
Intersection- The intersection of two sets A and B is the set consisting of all those elements that belong to both A and B and is denoted by A ∩ B.
Thus, A ∩ B = { x : x ∈ A and x ∈ B }.
Obviously, if A = { 1, 2, 3 } and B = { 2, 4, 6 }, then A ∩ B = { 2 }.
This is represented by the shaded part in (Fig. iv).
If B ⊂ A, then A ∩ B = B (Fig. v) and if A ⊂ B, then A ∩ B = A (Fig. vi).
It is worthwhile to remember in this case that two sets A and B are said to be comparable if either A ⊂ B or B ⊂ A.
Two sets A and B are said to be joint if they have at least one element in common. On the other hand, the two sets A and B are said to be disjoint, if they have no element in common i.e. if A ∩ B = Ø (Fig. vii).
For a finite family of sets A1, A2, …….An, the intersection of the sets if given by-
Difference of Sets- The difference of two sets A and B, written as A – B, is defined as the set consisting of all those elements of A which are not in B. On the other hand, B – A is defined to be the set consisting of all the elements of B which are not in A. Thus, if A = { 1, 2, 3, 4 } and B = { 2, 4, 6 } then A – B = { 1, 3 } and B – A = { 6 }.
Complement of a Set- Let ξ be the universal set and A ⊂ ξ. Then the complement of A with respect to ξ is the set consisting of the elements of ξ which are not in A and is denoted by A’ or Ac (Fig. viii).
| Thus, Ac = ξ – A or, Ac = { x : x ∈ ξ, x ∉ A } If ξ = { 1, 2, 3, 4, 5 } and A = { 2, 4 }, then Ac = { 1, 3, 5 } |
Laws of Complementation- The following properties are exhibited by the operation of complementation.
Symmetric Difference of Two Sets- The symmetric difference of two sets A and B is defined as (A – B) ∪ (B – A) and is denoted by A Δ B.
| Thus, if A = { 1, 2, 3, 4 } and B = { 2, 4, 6 }, then A – B = { 1, 3 } and B – A = { 6 } ∴ A Δ B = (A – B) ∪ (B – A) = { 1, 3 } ∪ { 6 } = { 1, 3, 6 } |
