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}

- A ∪ B = B ∪ A (commutative law)
- (A ∪ B) ∪ C = A ∪ (B ∪ C) (associative law)
- A ∪ ∅ = A (law of identity element)
- A ∪ A = A (idempotent law)
- U ∪ A = U (Uis the universal set)

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**.

- A ∩ B = B ∩ A (commutative law)
- (A ∩ B) ∩ C = A ∩ (B ∩ C) (associative law)
- ∅ ∩ A = ∅ ,U ∩ A = A
- A ∩ A = A (idempotent law)
- 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**.

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.

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

To solve math problems from Set Theory and learn math online, schedule a MathHelp session with our Math Tutors

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