Learn Miscellaneous Exercise 1 with Top Tutors

B = {2, 4, 6}, C = {2, 4, 6, 8…}, D = {6}.

A = {x: x ∈ R and x satisfies x² – 8x + 12 = 0}

2 and 6 are the only solutions of x² – 8x + 12 = 0.

∴ A = {2, 6}

B = {2, 4, 6}, C = {2, 4, 6, 8 …}, D = {6}

∴ D ⊂ A ⊂ B ⊂ C

Hence, A ⊂ B, A ⊂ C, B ⊂ C, D ⊂ A, D ⊂ B, D ⊂ C

(vi) If A ⊂ B and x ∉ B, then x ∉ A

(i) False

Let A = {1, 2} and B = {1, {1, 2}, {3}}

Now,

∴ A ∈ B

However,

(ii) False

Let

As A ⊂ B

B ∈ C

However,

(iii) True

Let A ⊂ B and B ⊂ C.

Let x ∈ A

∴ A ⊂ C

(iv) False

Let

Accordingly,and .

However, A ⊂ C

(v) False

Let A = {3, 5, 7} and B = {3, 4, 6}

Now, 5 ∈ A and A ⊄ B

However, 5 ∉ B

(vi) True

Let A ⊂ B and x ∉ B.

To show: x ∉ A

If possible, suppose x ∈ A.

Then, x ∈ B, which is a contradiction as x ∉ B

∴x ∉ A

Let, A, B and C be the sets such that and.

To show: B = C

Let x ∈ B

Case I

x ∈ A

Also, x ∈ B

∴

∴ x ∈ A and x ∈ C

∴ x ∈ C

∴ B ⊂ C

Similarly, we can show that C ⊂ B.

∴ B = C

(iii) A ∪ B = B (iv) A ∩ B = A

First, we have to show that (i) ⇔ (ii).

Let A ⊂ B

To show: A – B ≠ Φ

If possible, suppose A – B ≠ Φ

This means that there exists x ∈ A, x ≠ B, which is not possible as A ⊂ B.

∴ A – B = Φ

∴ A ⊂ B ⇒ A – B = Φ

Let A – B = Φ

To show: A ⊂ B

Let x ∈ A

Clearly, x ∈ B because if x ∉ B, then A – B ≠ Φ

∴ A – B = Φ ⇒ A ⊂ B

∴ (i) ⇔ (ii)

Let A ⊂ B

To show:

Clearly,

Let

Case I: x ∈ A

∴

Case II: x ∈ B

Then,

Conversely, let

Let x ∈ A

∴ A ⊂ B

Hence, (i) ⇔ (iii)

Now, we have to show that (i) ⇔ (iv).

Let A ⊂ B

Clearly

Let x ∈ A

We have to show that

As A ⊂ B, x ∈ B

∴

∴

Hence, A = A ∩ B

Conversely, suppose A ∩ B = A

Let x ∈ A

⇒

⇒ x ∈ A and x ∈ B

⇒ x ∈ B

∴ A ⊂ B

Hence, (i) ⇔ (iv).

Let A ⊂ B

To show: C – B ⊂ C – A

Let x ∈ C – B

⇒ x ∈ C and x ∉ B

⇒ x ∈ C and x ∉ A [A ⊂ B]

⇒ x ∈ C – A

∴ C – B ⊂ C – A