Proving A∪B=A∩B iff A=B with Boolean Algebra

[solakis1]

I want to prove:

\(\displaystyle A\cup B=A\cap B\Longleftrightarrow A=B\) Forall A,B sets

By using the axioms and theorems of the Boolean Algebra.

Any hints ??

 

[Evobeus]

I want to prove:

\(\displaystyle A\cup B=A\cap B\Longleftrightarrow A=B\) Forall A,B sets

By using the axioms and theorems of the Boolean Algebra.

Any hints ??

This answer is an answer based on Zermelo Fraenkel set theory axioms.
So, then consider the definition of "union" .
Given A and B , then \(\displaystyle A \cup B\) is a set C which contains all the elements that belong to A and all the elements that belong to B. In am more mathematical way C is a set such that \(\displaystyle d \in C\) implies and is implied by \(\displaystyle d \in A \) or \(\displaystyle d \in B\)
Now,
\(\displaystyle A\cup B=A\cap B\)

Considering the definition of intersection we get that every element of A is an element of B and every element of B is an element of A and thus from Axiom of Extensonality we get that A=B.