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Mathematics

Operation on Sets

Gk Scientist July 9, 2022 No Comments
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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).

Union of sets Venn Diagram - Operation on Sets

In the case of a finite family of sets A1, A2, A3, …….An, their union is given by-

union of finite family of sets - Operation on Sets

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).

Intersection of sets Venn Diagram - Operation on Sets

For a finite family of sets A1, A2, …….An, the intersection of the sets if given by-

intersetion of finite family of sets - Operation on Sets

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 }
Complement of a Set - Operation on Sets

Laws of Complementation- The following properties are exhibited by the operation of complementation.

Laws of Complementation - Operation on Sets

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 }

Sets Representation and TypesDual Nature of Radiations and Matter
Application of LogarithmsDerivation of de-Broglie equation
Pythagoras TheoremHeisenberg Uncertainty Principle
Linear Inequations in Two VariablesMagnetism– Tamil Board
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