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.

1 Answer
Feb 24, 2018

Proof:-" "Auu(BnnC)=(AuuB)nn(AuuC) A(BC)=(AB)(AC)

Let,
" "x in Auu(BnnC) xA(BC)

=>x in A vv x in (BnnC)xAx(BC)

=>x in A vv (x in B ^^ x in C)xA(xBxC)

=>(x in A vv x in B) ^^ (x in A vv x in C)(xAxB)(xAxC)

=>x in (A uu B) ^^ x in (A uu C)x(AB)x(AC)

=>x in (AuuB)nn(AuuC)x(AB)(AC)

  • x in Auu(BnnC)=>x in (AuuB)nn(AuuC)xA(BC)x(AB)(AC)

=>color(red)(Auu(BnnC)sube(AuuB)nn(AuuC)A(BC)(AB)(AC)

Let,
" "y in (AuuB)nn(AuuC) y(AB)(AC)

=>y in (A uu B) ^^ y in (A uu C)y(AB)y(AC)

=>(y in A vv y in B) ^^ (y in A vv y in C)(yAyB)(yAyC)

=>y in A vv (y in B ^^ y in C)yA(yByC)

=>y in A vv y in (BnnC)yAy(BC)

=>y in Auu(BnnC)yA(BC)

  • x in (AuuB)nn(AuuC)=>x in Auu(BnnC)x(AB)(AC)xA(BC)

=>color(red)((AuuB)nn(AuuC)subeAuu(BnnC)(AB)(AC)A(BC)

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...