Question

Which of the following statements are always true for any two sets A and B?

(a)

If A ⊆ B, then A ⊂ B.

(b)

If A ⊂ B, then A ⊆ B.

(c)

If A = B, then A ⊆ B.

(d)

If A = B, then A ⊂ B.

(e)

If A ⊂ B, then A ≠ B.

Answer 1

Answer:

If A ⊆ B, then A ⊂ B.

If A = B, then A ⊆ B.

If A = B, then A ⊂ B.

Explanation:

In set theory '⊂' is the symbol of proper subset and '⊆' is the symbol of subset of a set.

In option (a),

If A ⊆ B

⇒ A ⊂ B or A = B

Thus, If A ⊆ B, then A ⊂ B.

Option a is true.

(b) If A ⊂ B

⇒ A is the subset of B

That is, all elements of A are also the element of B,

But we can not say A = B

Thus, option b is not true.

(c) If A = B

⇒ A ⊂ B and A ⊃ B or A ⊆ B and B ⊆ A ( Because every set is the subset of itself )

⇒ A ⊆ B.

Option c is true.

(d) If A = B,

⇒ A ⊂ B

Option d is true.

(e) If A ⊂ B,

Then we can not say that,

A = B or A ≠ B

Thus, option e is not correct.

Answer 2

If A ⊆ B, then A ⊂ B.

If A = B, then A ⊆ B.

If A = B, then A ⊂ B.

Explanation:

In set theory '⊂' is the symbol of proper subset and '⊆' is the symbol of subset of a set.

In option (a),

If A ⊆ B

⇒ A ⊂ B or A = B

Thus, If A ⊆ B, then A ⊂ B.

Option a is true.

(b) If A ⊂ B

⇒ A is the subset of B

That is, all elements of A are also the element of B,

But we can not say A = B

Thus, option b is not true.

(c) If A = B

⇒ A ⊂ B and A ⊃ B or A ⊆ B and B ⊆ A ( Because every set is the subset of itself )

⇒ A ⊆ B.

Option c is true.

(d) If A = B,

⇒ A ⊂ B

Option d is true.

(e) If A ⊂ B,

Then we can not say that,

A = B or A ≠ B

Thus, option e is not correct.