# 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)# 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...