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Show that A ∪ B = A ∩ B implies A = BIn this question I have a doubt ie. what does this question mean. Should we prove A U B = A ∩ B given A = B or should we prove A = B given A ∪ B = A ∩ B​

Answer»

LET x∈A then x∈A∪B

SINCE , A∪B=A∩B  

x∈A∩B

So, x∈B  

i.e., if an element belongs to set A, then it must BELONG to set B ALSO.  

∴A⊂B.....(i)  

Similarly, if y∈B then, y∈A∪B  

Since A∪B=A∩B  

y∈A∩B  

So, y∈A  

∴ if y∈B  

then y∈A  

A⊂A.....(ii)  

from (i) and (ii)  

A⊂B and B⊂A  

thus A=B


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