DISJOINT SETS

Disjoint is one of the important operations on sets.

Let us discuss this operation in detail. 

Two sets X and Y are said to be disjoint if they do not have any common element. That is, X and Y are disjoint if

X n Y = Empty set

It is clear that n(A u B) = n(A) + n(B), if A and B are disjoint finite set.  

The Venn diagram given below illustrates the two disjoint sets X and Y. 

Example 1 :

Let A = {1, 3, 5, 6}, B = {0, 7}, find A n B. 

Solution :

A n B = {1, 3, 5, 6} u {0, 7}

A u B = {  } or empty set

Hence, A and B are disjoint sets.

Example 2 :

Let A = {0, 2, 7, 9}, B = {1, 3, 4, 7}, find A Δ B. When A and B are disjoint sets, verify A Δ B = A u B.

Solution :

A\B = A - B

= {0, 2, 7, 9} - {1, 3, 4, 7}

= {0, 2, 7, 9}

B\A = B - A

= {1, 3, 4, 7} - {0, 2, 7, 9}

= {1, 3, 4, 7}

A Δ B = (A\B) u (B\A)

A Δ B = {0, 2, 7, 9} u {1, 3, 4, 7}

A Δ B = {0, 2, 7, 9, 1, 3, 4, 7} ----(1)

A n B = {0, 2, 7, 9} n {1, 3, 4, 7}

A n B = { } ----(2)

A and B are disjoint sets.

A u B = {0, 2, 7, 9} u {1, 3, 4, 7}

A u B = {0, 2, 7, 9, 1, 3, 4, 7} ----(2)

From (1), (2) and (3), it is clear that if A are B disjoint sets, then

A Δ B = A u B

Related Pages

1. Union of sets

2. Intersection of sets

3. Set difference

4. Symmetric difference of two sets

5. Complement of a set

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