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3. Let A, B, and C be the sets such that A∪B=A∪C and A∩B=A∩C. Show that B=C.

Sets

Miscellaneous Exercise

http://ncert-solutions-class-11-chapter-1-sets-miscellaneous-exercise-question-3
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Class 11

Sets

Miscellaneous Exercise

1. Decide, among the following sets, which sets are subsets of one and another:

A={x : x∈R and x satisfy x2–8x+12=0}, B={2,4,6}, C={2,4,6,8, . . . }, D={6}.

2. In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.

(i) If x∈A and A∈B, then x∈B

(ii) If A⊂B and B∈C, then A∈C

(iii) If A⊂B and B⊂C, then A⊂C

(iv) If A⊄B and B⊄C, then A⊄C

(v) If x∈A and A⊄B, then x∈B

(vi) If A⊂B and x∉B, then x∉A

3. Let A, B, and C be the sets such that A∪B=A∪C and A∩B=A∩C. Show that B=C.

4. Show that the following four conditions are equivalent:

(i) A⊂B                 

(ii) A–B=\(\phi\)       

(iii) A∪B=B             

(iv) A∩B=A

5. Show that if AB, then C–B ⊂ C–A.

6. Show that for any sets A and B,

A=(A∩B)∪(A–B) and A∪(B–A)=(A∪B).

7. Using properties of sets, show that

(i) A∪(A∩B)=A         (ii) A∩(A∪B)=A.

8. Show that A∩B=A∩C need not imply B=C.

9. Let A and B be sets. If A∩X=B∩X=\(\phi\) and A∪X=B∪X for some set X, show that A=B.

(Hints A=A∩(A∪X), B=B∩(B∪X) and use Distributive law)

10. Find sets A, B and C such that A∩B, B∩C and A∩C are non-empty sets and A∩B∩C=\(\phi\).