Question

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.

Answer 1

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...