Prove that if A∩B = A∪B then A=B.
Having troubles figuring this one out.
Draw Venn diagram
Anyway to do it without drawing?
any element in one must be in the other.
any element not in one cannot be in the other :)
To prove that if A∩B = A∪B, then A = B, we need to show that A and B have the same elements.
To do this, we can prove two separate statements:
1. If A∩B = A∪B, then every element in A is also in B.
2. If A∩B = A∪B, then every element in B is also in A.
Let's start with statement 1:
Assume that A∩B = A∪B.
Take any arbitrary element x from A. Since x is in A, it must also be in A∪B.
Now, we know that A∩B = A∪B, which means that A∩B is a subset of A∪B. Therefore, x must also be in A∩B.
However, x is in both A and A∩B. Since A∩B is a subset of A, x belongs to A∩B implies that x belongs to A. Thus, every element in A is also in B.
Now, let's move on to statement 2:
Assume that A∩B = A∪B.
Take any arbitrary element y from B. Since y is in B, it must also be in A∪B.
Again, A∩B is a subset of A∪B. Therefore, y must also be in A∩B.
However, y is in both B and A∩B. Since A∩B is a subset of B, y belongs to A∩B implies that y belongs to B. Thus, every element in B is also in A.
Since we have shown that every element in A is also in B and every element in B is also in A, we can conclude that A = B.
Therefore, if A∩B = A∪B, then A = B.