**Disjoint sets**

Two sets are **disjoint** if they have no elements in common, i.e., if their intersection is the empty set.

⛲ Example 1. Is there a disjoint pair from among *C*={1,2,3,4}, *D*={2,4,6,8} and *E*={3,5,7,9}?

• *D*∩*E*=Ø, so *D* and *E* are disjoint.

*C* and *D* have elements in Common, so they are not disjoint. Also, *C* and *E* are not disjoint.

**Disjoint sets**

Two sets are called disjoint if they have no elements in common. For example:

*S*={2,4,6,8} and

*T*={1,3,5,7} are disjoint.

Another way to define disjoint sets is to say that their intersection is the empty set. Two sets

*A*and

*B*are disjoint if

*A*∩

*B*=Ø.

In the example above,

*S*∩

*T*=Ø because no number lies in both sets.

**Representing disjoint sets on a Venn diagram**

When we know that two sets are disjoint,we represent them by circles that do not intersect. For example,let

*P*={0,1,2,3} and

*Q*={8,9,10}

Then

*P*and

*Q*are disjoint,as illustrated in the Venn diagram below.

[Definition] Two sets are said to be disjoint if they have no element in common.

⛲ Ex2. The sets, *F*={0,4,7,9} and *G*={3,6,10} are disjoint.

⛲ Ex3. State whether each of the following statement is true or false. Justify your answer.

(i) {2,3,4,5} and {3,6} are disjoint sets.

(ii) {a,e,i,o,u } and {a,b,c,d} are disjoint sets.

(iii) {2,6,10,14} and {3,7,11,15} are disjoint sets.

(iv) {2,6,10} and {3,7,11} are disjoint sets.

✍ Solution:

(i) False

As 3∈{2,3,4,5}, 3∈{3,6}

⇒{2,3,4,5}∩{3,6}={3}

(ii) False

As a∈{a,e,i,0,u}, a∈{a,b,c,d}

⇒{a,e,i,o,u}∩{a,b,c,d}={a}

(iii) True

As {2,6,10,14}∩{3,7,11,15}=Ø

(iv) True

As {2,6,10}∩{3,7,11}=Ø

⛲ Ex4. State whether each of the following statement is true or false.

(i) *H*={2,4,6,8} and *I*={1,3,5} are disjoint sets.

(ii) *J*={a,e,i,o,u} and *K*={a,b,c,d} are disjoint sets.

✍ Solution:

(i) We have,*H*={2,4,6,8} and *I*={1,3,5}

Now, *H*∩*I*={2,4,6,8}∩{1,3,5}=Ø.

Therefore, *H* and *I* are disjoint sets. Hence, given statement is true.

(ii) We have, *J*={a,e,i,o,u} and *K*={a,b,c,d} Now, *J*∩*K*={a}. ∴ *J*∩*K*≠Ø.

Therefore, *J* and *K* are not disjoint sets. Hence, given statement is false.

⛲ Ex5. Which of the following pairs of sets are disjoint

(i) {1,2,3,4} and {*x*:*x* is a natural number and 4≤*x*≤6}

(ii) {a,e,i,o,u} and {c,d,e,f}

(iii) {*x*:*x* is an even integer} and {*x*:*x* is an odd integer}

✍ Solution:

(i) {1,2,3,4}

{*x*:*x* is a natural number and 4≤*x*≤6}={4,5,6}.

Now, {1,2,3,4}∩{4,5,6}={4}

Therefore,this pair of sets is not disjoint.

(ii) {a,e,i,o,u}∩(c,d,e,f}={e}

Therefore, {a,e,i,o,u} and (c,d,e,f} are not disjoint.

(iii) {*x*:*x* is an even integer}∩{*x*:*x* is an odd integer}=Ø

Therefore, this pair of sets is disjoint.

⛲ Ex6. Which of the following pairs of sets are disjoint?

(i) *L*={1,2,3,4,5,6} and *M*={*x*:*x* is a natural number and 4≤*x*≤6}

(ii) *N*={*x*:*x* is the boys of your school}, *O*={*x*:*x* is the girls of your school}

🌟 Firstly, convert all the sets in roster form, if it is not given in that. Then use the condition for disjoint sets i.e., *A*∩*B*=Ø.

✍ Solution:

(i) Given, *L*={1,2,3,4,5,6} and *M*={4,5,6}

∴ *L*∩*M*={1,2,3,4,5,6}∩{4,5,6}={4,5,6}≠Ø

Hence, this pair of sets is not disjoint.

(ii) Here, *N*={b_{1},b_{2},…} and *O*={g_{1},g_{2},…},

where b_{1},b_{2},…, are the boys and g_{1},g_{2},…, are the girls of school.

Clearly, *N*∩*O*=Ø

Hence, this pair of set is disjoint set.