Math Examples of Basic Operations on Sets

Union of Sets, represented as A U B. Word Union means adding together. When we have the two Sets A and B and the resultant Set is A U B which has all the elements of set A and set B. It is read as A Union B. Let A = 1, 2, 3, 4 and B = 4, 5, 6 then we can say that
A U B = 1, 2, 3, 4, 5, 6 which includes all the elements of set A and set B.

The Basic Operations on the two Sets A and B can be given as follows:
1.       Union of Sets
2.       Intersection of sets
3.       Complement of the Set
4.       Cartesian Product of sets

Intersection of Sets, represented by A ∩ B. Word Intersection means the elements of the two Sets which are common. So if we write Let A = 1, 2, 3, 4 and B = 4, 5, 6 then we can say that
A ∩ B = 4 which includes all the common elements of Set A and set B. Here we observe that only element 4 is common in both the sets A and B.

If we have two Sets ‘A’ and ‘B’ then Relative complement means all the elements of ‘A’ which are not in Set ‘B’. It is expressed by ‘AB’ or we write it as A- B. Let us consider an example such that A= 1, 2, 3, 4 and B= table, Chair. Here we write AB = 1, 2, 3, 4 where it has all the elements of ‘A’ which are not in ‘B’. In another Example if A= 1, 2, 3, 4 and B= 1, 2. Then we write AB = 3, 4 which shows 3, 4 are the elements of ‘A’ which are not in ‘B’. Similarly if ‘A’ and ‘B’ have all elements common, then AB = φ (Null set).

Complement of a Set A:  Let U be a universal set and let A be any set, then the complement of set A is A’ which is represented by U – A. Eg: Let U = 1, 2, 3, 4, 5, 6 and A = 1, 2, 3, then A’ = U – A
So A’ = 4, 5, 6 which means that complement of A = all the elements of universal set U which are not in set A.

If ‘A’ and ‘B’ are any two Sets then A * B is represents the Cartesian Product, which includes Set of all ordered pairs (a, b) in such a way that a is the member of set ‘A’ and ‘B’ is the member of set ‘B’. For example: If A= 1, 2, and B = 3, 4, then A * B = 1, 3, 1, 4 2, 3, 2, 4 . So it has 4 elements in this set.

Let A and B be any two Sets, then we write:
                A Union B = A U B,
               A Intersection B = A ∩ B,
            Complement of A = A’ and for Cartesian product of Set A and set B, we have A*B.

Let us take two Sets A= 1, 2, 3, 4, 5, 6, and  B = 4, 5, 6, 7, 8
   Now if we write A U B, it means adding up all the elements of Set A and Set B.
So A U B = 1, 2, 3, 4, 5, 6, 7, 8
 Similarly if we write B U A, it means adding up all the elements of B and A set
So B U A = 1, 2, 3, 4, 5, 6, 7, 8
 From above we conclude A U B = B U A.

We know that a complement of ‘E’ is represented by E’.
So E’ = U – E,
Now we see that if from the Set of integers, if all even Numbers are taken out , we are left with the set of Odd Numbers. So we can say that the complement of Even Numbers is the set of odd numbers.
Or we can say E’ = O (i.e. a set of odd numbers). 

A ∩ φ = A is not a True statement. As when we write A ∩ φ, it means the intersecting elements of Set ‘A’ and a null set φ.  Let if A = 1, 3, 5 and we know that null set has no element in it. Then the Intersection of two such Sets is always a null set, because intersection means the common elements of the two Sets. So we come to a conclusion that A ∩ φ = φ.