Let S be a given set and A / subset S and B / subset S. Prove that A / subset B / Leftrightarrow S/A / supset S/B.

Here,

S is a set so that A ⊂ S and B ⊂ S,

We have to prove that,

A ⊂ B ⇔ S/A ⊂ S/B.

Suppose,

A ⊂ B

Also, let x ∈ S/B, where, x is an arbitrary,

⇒ x ∈ S but x ∉ B

⇒ x ∈ S but x ∉ A (∵ A ⊂ B)

⇒ x ∈ S/A

⇒ x ∈ S/B ⇒ x ∈ S/A

⇒ S/A ⊂ S/B.

Conversely,

Suppose, S/A ⊂ S/B,

Let y ∈ B, where y is an arbitrary,

⇒ y ∉ S - B

⇒ y ∉ S - A (∵ S/A ⊂ S/B)

⇒ y ∈ A but y ∉ S

⇒ y ∈ B ⇒ y ∈ A

⇒ A ⊂ B

Hence, proved ...