For any two sets A and B , prove that A ∪ B = A ∩ B if and only if A = B .

First, let A = B then, A ∪ B = A and A ∩ B = A ⇒ A ∪ B = A ∩ B Thus, A = B ⇒ A ∪ B = A ∩ B . . . . . . . . . ( i ) Conversely, let A ∪ B = A ∩ B . Then we have to prove that A = B for this, let x ∈ A ⇒ x ∈ A ∪ B ⇒ x ∈ A ∩ B [ ∵ A ∪ B = A ∩ B ] ⇒ x ∈ A and x ∈ B ⇒ x ∈ B ∴ A ⊂ B . . . . . . . . . ( i i ) y ∈ B ⇒ y ∈ A ∪ B ⇒ y ∈ A ∩ B ⇒ y ∈ A and y ∈ B ⇒ y ∈ A ∴ B ⊂ A . . . . . . . . . . ( i i i ) From ( i i ) and ( i i i ) , we get A = B . . . . . ( i v ) Thus, A ∪ B = A ∩ B ⇒ A = B From ( i ) and ( i v ) , we obtain A ∪ B = A ∩ B ⇔ A = B

For any two sets A and B , prove that A ∪ B = A ∩ B if and only if A = B .

First, let A = B then, A ∪ B = A and A ∩ B = A ⇒ A ∪ B = A ∩ B Thus, A = B ⇒ A ∪ B = A ∩ B . . . . . . . . . ( i ) Conversely, let A ∪ B = A ∩ B . Then we have to prove that A = B for this, let x ∈ A ⇒ x ∈ A ∪ B ⇒ x ∈ A ∩ B [ ∵ A ∪ B = A ∩ B ] ⇒ x ∈ A and x ∈ B ⇒ x ∈ B ∴ A ⊂ B . . . . . . . . . ( i i ) y ∈ B ⇒ y ∈ A ∪ B ⇒ y ∈ A ∩ B ⇒ y ∈ A and y ∈ B ⇒ y ∈ A ∴ B ⊂ A . . . . . . . . . . ( i i i ) From ( i i ) and ( i i i ) , we get A = B . . . . . ( i v ) Thus, A ∪ B = A ∩ B ⇒ A = B From ( i ) and ( i v ) , we obtain A ∪ B = A ∩ B ⇔ A = B

E M B I B E

Golden MATHEMATICS CLASS 11>Chapter 1 - Sets>SELF EVALUATION TEST>Q 5

HARD

11th CBSE

IMPORTANT

Earn 100

For any two sets $A$ and $B$ , prove that $A \cup B = A \cap B$ if and only if $A = B$ .

Important Questions on Sets

EASY

11th CBSE

IMPORTANT

Golden MATHEMATICS CLASS 11>Chapter 1 - Sets>SELF EVALUATION TEST>Q 6

If

a \in N

such that

a N = {a x : x \in N}

and

3 N \cap 7 N = k N

, then find the value of

k

.

HARD

11th CBSE

IMPORTANT

Golden MATHEMATICS CLASS 11>Chapter 1 - Sets>SELF EVALUATION TEST>Q 7

In a group of

50

people,

35

speak Hindi,

25

speak both Hindi and English and all speak at least one of the two languages. How many people speak only English but not Hindi? How many people speak English?

EASY

11th CBSE

IMPORTANT

Golden MATHEMATICS CLASS 11>Chapter 1 - Sets>SELF EVALUATION TEST>Q 8

If