Union of Sets

Let A and B be two non–empty sets. The union of A and B is the set which consists of all the elements of A and all the elements of B and the common elements of A and B are taken only once.

We denote union of two sets by the symbol ‘U’ and write as A U B and usually read as ‘A union B’.
 

Example: Let A = {2, 4, 6, 8, 10} and B = {1, 3, 5, 7, 9} be two sets
So, A U B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

Thus, we can define the union of two sets as:
The union of two sets A and B is the set C which consists of all those elements which are either in A or in B (including those which are in both)

A U B = {x: x ε A or x ε B}

Properties of the Operation of Union

1) A U B = B U A (Commutative law)

2) (A U B) U C = A U (B U C) (Associative law)

3) A U Ф = A (Law of Identity element, Ф is the identity of U)

4) A U A = A (Idempotent law)

5) U U A = U (Law of U)

Intersection of Sets

Let A and b be two non–empty sets. The intersection of sets A and B is the set of all elements which are common to both A and B.

We denote intersection of two sets by the symbol ‘∩’ and write as A ∩ B and usually read as ‘A intersection B’.

 

Example: Let A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and B = {2, 3, 5, 7} be two sets

So, A ∩ B = {2, 3, 5, 7}

 

From the above discussion, the intersection of two sets A and B is the set of all those elements which belong to both A and B.

A ∩ B = {x: x ε A and x ε B}

Properties of the Operation of Intersection

1) A ∩ B = B ∩ A (Commutative law)

2) (A ∩ B) ∩ C = A ∩ (B ∩ C) (Associative law)

3) Ф ∩ A = Ф, U ∩ A = A (Law of Ф and U)

4) A ∩ A = A

5) A ∩ (B ∩ C) = (A ∩ B) ∩  (A ∩ C) (Distributive law)
 

Difference of Sets

If A and B are two non–empty sets then the difference of the sets A and B in the same order is the set of elements which belong to A but not to B.
We write it as, A – B and read as A minus B.
Example: Let A = {2, 3, 5, 6, 9} and B = {1, 2, 4, 6, 9}, find A – B and B – A.
A – B = {3, 5}, since the elements 3, 5 belong to A but not to B.
B – A = {1, 4}, since the elements 1, 4 belong to B but not to A.

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