i) AU B = B U A

ii) A ∩ B = B ∩ A.

Hint: As you are given choice to select two sets write any two sets A

and B and try to prove the relations. For example:

Solution: A = {1, 2, 3, 4, 5}

B = {3, 4, 5, 6, 7}

i) AU B = {1, 2, 3, 4, 5} U {3, 4, 5, 6, 7}

= {1, 2, 3, 4, 5, 6, 7}

BUA = {1, 2, 3, 4, 5, 6, 7}

A U B = B U A

ii) A ∩ B = {3, 4, 5}

B ∩ A = {3, 4, 5}

A ∩ B = B ∩ A

Q: From the given venn diagram, find

1) X U Y 3) X − Y

2) X ∩ Y 4) Y − X

What can you infer about X . Y, X n Y and X - Y?

Hint: Write the sets in roster form and answer the given questions. As

X Y, X - Y will be the empty set.

Solution: X = {2, 4}

Y = {1, 2, 3, 4, 5}

1) X . Y = {1, 2, 3, 4, 5} = Y

2) X n Y = {2, 4} = X

3) X - Y = {2, 4} - {1, 2, 3, 4, 5}= { }

4) Y - X = {1, 2, 3, 4, 5} - {2, 4} = {1, 3, 5}

X - Y is an empty set as X Y.

Q: Sneha has taken two sets of her choice and observed that n (A . B)

= n (A) + n (B). Jhansi argued that

n (A . B) = n (A) + n (B) - n (A n B). Who is correct? Give your reasons.

Hint: Jhansi's statement is correct for any two sets A and B. And

Sneha's statement is correct when the sets A and B are disjoint.

Solution: If the sets A and B are disjoint n (A n B) = 0

n (A . B) = n (A) + n(B)

This shows that Sneha has considered two disjoint sets.

But, for any two sets A and B

n (A . B) = n (A) + n (B) - n (A n B) is always true.

Jhansi's statement is true for any two sets A and B.

And Sneha's statement is conditionally true i.e., for only two disjoint sets.

Q: If n (A − B) = 10 and n (B − A) = 20 and n (A U B) = 45 then find n (A ∩ B).

Hint: This problem can be solved easily using a venn diagram.

Solution:

n (A U B) = n (A − B) + n (A ∩ B) + n (B − A)

45 = 10 + n (A ∩ B) + 20

n (A ∩ B) = 45 − 30 = 15

Q: Which of the following sets are infinite. Try to impose a condition

to make them finite.

i) A = {2, 4, 6, 8, ...}

ii) B = {a, e, i, o, u}

iii) C = {..., −3, −2, −1, 0, 1, 2, 3, ...}

Hint: You have many options to impose conditions to make the infinite

Sets A and C finite.

Solution:

Sets A and C are infinite sets as the elements of these sets are infinite.

i) A = {2, 4, 6, 8, ...}

The elements of Set A are multiples of 2. If the elements of Set A are

considered as the multiples of 2 less than 10 then it becomes finite.

ii) C = {..., −3, −2, −1, 0, 1, 2, 3, ....}

The elements of Set C are integers. If the elements of Set C are taken

as the integers that lie between −5 and 5 it becomes finite.

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