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
Proof:-
" "Auu(BnnC)=(AuuB)nn(AuuC) A∪(B∩C)=(A∪B)∩(A∪C) Let,
" "x in Auu(BnnC) x∈A∪(B∩C)
=>x in A vv x in (BnnC)⇒x∈A∨x∈(B∩C)
=>x in A vv (x in B ^^ x in C)⇒x∈A∨(x∈B∧x∈C)
=>(x in A vv x in B) ^^ (x in A vv x in C)⇒(x∈A∨x∈B)∧(x∈A∨x∈C)
=>x in (A uu B) ^^ x in (A uu C)⇒x∈(A∪B)∧x∈(A∪C)
=>x in (AuuB)nn(AuuC)⇒x∈(A∪B)∩(A∪C)
x in Auu(BnnC)=>x in (AuuB)nn(AuuC)x∈A∪(B∩C)⇒x∈(A∪B)∩(A∪C)
=>color(red)(Auu(BnnC)sube(AuuB)nn(AuuC)⇒A∪(B∩C)⊆(A∪B)∩(A∪C) Let,
" "y in (AuuB)nn(AuuC) y∈(A∪B)∩(A∪C)
=>y in (A uu B) ^^ y in (A uu C)⇒y∈(A∪B)∧y∈(A∪C)
=>(y in A vv y in B) ^^ (y in A vv y in C)⇒(y∈A∨y∈B)∧(y∈A∨y∈C)
=>y in A vv (y in B ^^ y in C)⇒y∈A∨(y∈B∧y∈C)
=>y in A vv y in (BnnC)⇒y∈A∨y∈(B∩C)
=>y in Auu(BnnC)⇒y∈A∪(B∩C)
x in (AuuB)nn(AuuC)=>x in Auu(BnnC)x∈(A∪B)∩(A∪C)⇒x∈A∪(B∩C)
=>color(red)((AuuB)nn(AuuC)subeAuu(BnnC)⇒(A∪B)∩(A∪C)⊆A∪(B∩C)
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...
