Show that the following four conditions are equivalent:
(i) $A \subset B$ (ii) $A - B = ϕ$ (iii) $A \cup B = B$ (iv) $A \cap B = A$

Question

Show that the following four conditions are equivalent:(i) A⊂B (ii) A−B=ϕ (iii) A∪B=B (iv) A∩B=A

Filo · Accepted Answer

First, we have to show that (i) ⇔ (ii).(i)⇒ (ii).Let A⊂BTo show: A−B=ϕIf possible, suppose A−B=ϕThis means that there exists x∈A,x∈/B, which is not possible as A⊂B.∴A−B=ϕ∴A⊂B⇒A−B=ϕ(ii)⇒ (i).Let A−B=ϕTo show: A⊂BLet xϵAClearly, xϵB because if xϵB, then A−B=ϕ∴A−B=ϕ⇒A⊂BHence, (ii)⇔(i)(i)⇒ (iii).Let A⊂BTo show: A∪B=BClearly, B⊂A∪BLet xϵA∪B⇒xϵA or xϵBCase I: xϵA⇒xϵB      [∵A⊂B]∴A∪B⊂BCase II: xϵBThen A∪B⊂BSo, A∪B=B(iii)⇒ (i).Conversely, let A∪B=BTo show : A⊂BLet xϵA⇒xϵA∪B   [∵A⊂A∪B]⇒xϵB     [∵A∪B=B]∴A⊂BHence, (iii)⇔(i)Now, we have to show that (i)⇔(iv).Let A⊂BClearly A∩B⊂ALet xϵAWe have to show that xϵA∩BAs A⊂B,xϵB∴xϵA∩B∴A⊂A∩BHence, A=A∩BConversely, suppose A∩B=ALet xϵA⇒xϵA∩B⇒xϵA and xϵB⇒xϵB∴A⊂BHence, (i)⇔(iv).

Question Text	Show that the following four conditions are equivalent: (i) $A \subset B$ (ii) $A - B = ϕ$ (iii) $A \cup B = B$ (iv) $A \cap B = A$
Answer Type	Text solution:1
Upvotes	150

Show that the following four conditions are equivalent:
(i) $A \subset B$ (ii) $A - B = ϕ$ (iii) $A \cup B = B$ (iv) $A \cap B = A$

Text solutionVerified

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