garnerashton5335 garnerashton5335
  • 02-10-2019
  • Mathematics
contestada

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.

Respuesta :

slicergiza slicergiza
  • 04-10-2019

Answer:

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...

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