#### Question

Show that the following four conditions are equivalent:

**(i)** A ⊂ B

**(ii)** A – B = Φ

**(iii)** A ∪ B = B

**(iv)** A ∩ B = A

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#### Solution

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

Conversely, suppose A ∩ B = A

Let *x* ∈ A

⇒ x ∈ A ∩ B

⇒ *x* ∈ A and *x* ∈ B

⇒ *x *∈ B

∴ A ⊂ B

Hence, (i) ⇔ (iv).

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#### Reference Material

Solution for question: Show that the Following Four Conditions Are Equivalent:A Subset B, a Difference B = φ, a Union B = B And a Intersection B = a concept: Subsets. For the courses CBSE (Arts), CBSE (Science), CBSE (Commerce)