Question

If A and B are subsets of the universal set U, then which of the following is/are true?

• A. A ⊂ (A ∪ B)
• B. A ⊂ B ⇔ (A ∪ B) = B
• C. (A ∩ B) ⊂ A
• D. A and B cannot be subsets of each other.

Answer 1

Answer: (A), (B), (C)

The correct options are
A A ⊂ (A ∪ B)
B A ⊂ B ⇔ (A ∪ B) = B
C (A ∩ B) ⊂ A

A ⊂ (A ∪ B) is true as A ∪ B is a collection of all elements of A and B.
$\therefore$ A is a subset of A ∪ B.

A ⊂ B ⇔ (A ∪ B) = B is true. If A is a subset of B, then all elements of A are elements of B also.
$\therefore$ (A ∪ B) = B.
If (A ∪ B) = B, this means all the elements of set A and set B are in set B.
$\therefore$ All the elements of set A are also elements of set B. Hence, A ⊂ B.

(A ∩ B) ⊂ A is true as A ∩ B is a collection of elements common to both set A and set B. Hence, elements in A ∩ B will always be present in both set A and set B.
$\therefore$ (A ∩ B) is a subset of A.

A and B can be subsets of each other when they are equal.
$\therefore$ This statement is incorrect.