Question

How to prove this? 6) For sets A,B,C prove A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) by showing Left side ⊆ Right side and Right side ⊆ Left side.

Answer 1

Proof:-`" "Auu(BnnC)=(AuuB)nn(AuuC)`

Let,
`" "x in Auu(BnnC)`

`=>x in A vv x in (BnnC)`

`=>x in A vv (x in B ^^ x in C)`

`=>(x in A vv x in B) ^^ (x in A vv x in C)`

`=>x in (A uu B) ^^ x in (A uu C)`

`=>x in (AuuB)nn(AuuC)`

• `x in Auu(BnnC)=>x in (AuuB)nn(AuuC)`

`=>color(red)(Auu(BnnC)sube(AuuB)nn(AuuC)`

Let,
`" "y in (AuuB)nn(AuuC)`

`=>y in (A uu B) ^^ y in (A uu C)`

`=>(y in A vv y in B) ^^ (y in A vv y in C)`

`=>y in A vv (y in B ^^ y in C)`

`=>y in A vv y in (BnnC)`

`=>y in Auu(BnnC)`

• `x in (AuuB)nn(AuuC)=>x in Auu(BnnC)`

`=>color(red)((AuuB)nn(AuuC)subeAuu(BnnC)`

From the both red part , we get by using the rule of equal set,

`color(red)(ul(bar(|color(green)(Auu(BnnC)=(AuuB)nn(AuuC)))|`

Hope it helps...
Thank you...