Show that A⋃B = A∩B implies A=B.
Get the complete, step-by-step math solution for: "Show that A B = A B implies A = B.". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Assume the given condition
We start by assuming the given condition that the union of sets A and B is equal to their intersection. Our goal is to prove that this condition implies that set A must be equal to set B.
Step 2: Consider an element in A
To prove that A=B, we need to show that A⊆B and B⊆A. Let's start by assuming an arbitrary element x belongs to set A.
Step 3: Use the union property
If an element x is in set A, then by the definition of union, x must also be in the union of A and B, denoted as A⋃B.
Step 4: Apply the given condition
From our initial assumption, we know that A⋃B=A∩B. Therefore, if x is in A⋃B, it must also be in A∩B.
Step 5: Deduce A is a subset of B
If x is in the intersection of A and B, it means x is in both A and B. Since we started with x∈A and concluded x∈B, this shows that every element of A is also an element of B, meaning A is a subset of B.
Step 6: Deduce B is a subset of A
By following the same logic, if we assume an arbitrary element y is in B, then y must be in A⋃B. Using the given condition, y must also be in A∩B, which implies y is in A. Thus, B is a subset of A.
Step 7: Conclusion
Since we have shown that A is a subset of B and B is a subset of A, by the definition of set equality, it must be that A=B.