Operations on Set

Union Set

The union of two sets A and B is the set which consists of all those elements which are either in A or in B (including those which are in both).

The symbol for union of sets is ∪

In symbols the union of two sets is written as :

A ∪ B= { x ∶ x ∈ A or x ∈ B}

Let A = {4,5,6} and B = { 4,5, 6, 3, 8, 9}

Then the union of two sets A and B will be A ∪ B = { 3,4,5,6,8,9}

The Union of sets has following properties:

  1. A ∪ B = B ∪ A (commutative law)
  2. (A ∪ B) ∪ C = A ∪ (B ∪ C) (associative law)
  3. A ∪ ∅ = A (law of identity element)
  4. A ∪ A = A (idempotent law)
  5. U ∪ A = U (Uis the universal set)

Intersection of Sets

The intersection of two sets A and b is the set of all those elements which belong to both A and B.

The symbol for intersection is ∩

The notation of intersection of sets is : A ∩ B={ x ∶ x ∈ A and x ∈ B}

Example of intersection of two sets can be : Let A = { 1, 2, 3} and B = { 2, 3, 7, 8 } then A ∩ B = { 2, 3}

If A and B are two sets such as A ∩ B = ∅ then A and B are called disjoint sets.

Intersection of sets has the following properties:

  1. A ∩ B = B ∩ A (commutative law)
  2. (A ∩ B) ∩ C = A ∩ (B ∩ C) (associative law)
  3. ∅ ∩ A = ∅ ,U ∩ A = A
  4. A ∩ A = A (idempotent law)
  5. A ∩(B ∪ C) = (A ∩ B) ∪ (A ∩ C) (distributive law)

The difference A – B is the set of elements which belong to A and not to B .

A - B = {x : x ∈ A and x ∈ B}

The difference set B – A is the set of all those elements of B which are not in A.

e.g. A = { 2, 4, 6, 8, 11} and B = { 1, 11, 13}

then A – B = { 2,4,6,8}

B – A = { 1, 13}

It is obvious that A – B ≠B – A

The sets A - B,A ∩ B and B - A are mutually disjoint sets.

Symmetric Difference of Sets

The symmetric difference of two sets is the set of elements which are in either of the two sets but do not belong to the intersection of the set.

Complement of a set

Let U be the universal set and M ⊂ U . Then the complement of set M is the set of all elements of U which are not the elements of M. The compliment of set M is denoted as M’. Thus

M'={ x ∈ U such that x ∈ M}

Obviously M' = U - M

For any two sets A and B, we can prove the following properties of complement of a set:

1. (A' )' = A , or, the complement of the complement of a set is the set itself.

2. (A ∪ B)' = A'∩ B' and (A ∩ B)' = A' ∪ B' : This is also known as the De Morgan’s law in set theory which says that union and intersection interchange under complementation. Thus the complement of the union of two sets A and B is the intersection of the two complements and the complement of the intersection of two sets A and B is the union of their complements.

3. A ∪ A' = U or the union of a set with it’s complement is the universal set

4. A ∩ A' = ∅ or the intersection of a set with it’s complement is a null set.

5. ∅' = U and U' = ∅ or the universal set and the null set are complements of each other

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