Prove or Disprove: if the complement of A is a subset of B then A union B= U
This is as far as I got.
The complement of A is anything that is NOT A.
If everything that is NOT A IS a subset of B then that does NOT necessarily mean that A union B=U.
Let x be an element in the complement of set A.
If x E of the complement of set A, and the complement of A is the subset of B then A also must be the subset of B. Because A is the subset of B then A union B= B it is NOT EQUAL to U. However, the complement of B union B would equal the universe.
Okay I THINK I got it right, can someone please take a look?
Prove or Disprove: If the complement of A is the subset of B then A union B= Universe
TRUE.
Proof By contradiction:
Suppose not. That is, suppose that if the complement of A is the subset of B then A union B DOES NOT equal U.
Let,
Complement of A= {1,2,3}
A= {4,5,6,7}
B={1,2,3,4,5}
U={1,2,3,4,5,6,7}
Then,
The complement of A subset B= {1,2,3}
A union B= {1,2,3,4,5,6,7} = U
This is a contradiction b/c we supposed that if the complement of A is the subset of B then A union B DOES NOT equal the universe.
However, the original statement is TRUE b/c it IS the case that when the complement of A is the subset of B then A union B = U. **END OF PROOF***
Yes I read it, Thank you for taking the time to respond. However, while I was waiting for someone to respond I came up with my own proof, I don't doubt your proof is correct. I just wanted to see if my proof is incorrect Or maybe both methods are valid methods to prove this?
You mean your argument in your post #3? No that argument makes no sense. I'll explain.
First, though, yes the statement in question is provable (but your approach itself is not correct).
Let '\' stand for the binary complement operation.
And since U is the universe, we suppose:
A subset of U
and
B subset of U.
Theorem:
If A subset of U
and
B subset of U
and
U\A subset of B,
then AuB = U.
Proof (essentially the same as poster Plato's correct argument):
Suppose x in AuB. So x in A or x in B. But since both A and B are subsets of U, we have x in U. So AuB subset of U.
Now suppose x in U. If x in A then, since A subset of U, we have x in U. And if x not in A then x in U\A, which is a subset of B, so x in B. So since x either in A or not in A, we have x in A or x in B. So X in AuB. So U subset of AuB.
So AuB subset of U, and U subset of AuB, so AuB = U.
Done.
Now, here's what's wrong with your approach. What you did is show that for PARTICULAR sets A, B, and U, we have the desired result. But that's not at all how proof by contradiction works in this case. In this case, proof by contradiction is to assume is that there exist sets A, B, and U for which the result fails and to derive a contradiction. The fact that the result does hold for some particular A, B, and U in no way proves that there isn't also some A, B, and U for which the result fails, thus you've not derived any contradiction. All you have is that the result holds for some A, B, and U along with the assumption that it fails for some A, B, and U. But that is not a contradiction.
Anyway, we don't need proof by contradiction in this case. All we need is the straightforward approach to showing to sets equal by showing each is a subset of the other, just as the poster Plato first gave it to you.