Difference of Two Sets

How
to find the difference of two sets?

If A and B are two sets, then their difference is given by A – B or B – A. 

• If A = {2, 3, 4} and B = {4, 5, 6} 

A – B means elements of A which are not the elements of B. 

i.e., in the above example A – B = {2, 3} 

In general, B – A = {x : x ∈ B, and x ∉ A} 

• If A and B are disjoint sets, then A – B = A and B – A = B 

Solved examples to find the
difference of two sets:

1. A = {1, 2, 3} and B = {4, 5, 6}. 

Find
the difference between the two sets:

(i)
A and B

(ii)
B and A

Solution:

The two sets are disjoint as they do not have any elements in common. 

(i) A – B = {1, 2, 3} = A

(ii)
B – A = {4, 5, 6} = B 

2. Let A = {a, b, c, d, e, f} and B = {b, d, f, g}.

Find
the difference between the two sets:

(i)
A and B

(ii)
B and A

Solution:

(i) A – B = {a, c, e}

Therefore,
the elements a, c, e belong to A but not to B 

(ii) B – A = {g) 

Therefore,
the element g belongs to B but not A. 

3. Given three sets P, Q and R such that:

P = {x : x is a natural number between 10
and 16},

Q = {y : y is a even number between 8 and
20} and

R =
{7, 9, 11, 14, 18, 20}

(i) Find the difference of two sets P and Q

(ii) Find Q – R

(iii) Find R – P

(iv) Find Q – P

Solution:

According to the given statements:

P = {11, 12, 13, 14, 15}

Q = {10, 12, 14, 16, 18}

R = {7, 9, 11, 14, 18, 20}

(i) P – Q = {Those elements of set P which
are not in set Q}

=
{11, 13, 15}

(ii) Q – R = {Those elements of set Q not
belonging to set R}

=
{10, 12, 16}

(iii) R – P = {Those elements of set R
which are not in set P}

=
{7, 9, 18, 20}

(iv) Q – P = {Those elements of set Q not
belonging to set P}

= {10, 16, 18}