Therefore, B is not a subset of UProving A n B = U iff A = U and B = U

[mbcsantin]

Homework Statement

Prove the following: For any sets A, B, C in a universe U:
A n B = Universe iff A = Universe and B = Universe

Homework Equations

none.

The Attempt at a Solution

I tried to do the questions but I am just not sure if i did it right. id appreciate if you can check my work and let me know what changes i have to make. thanks

the symbol "n" means "intersect"
U for Union

Suppose A n B = U and suppose that A is a proper subset of U then
x is an element of B but
x is not an element of A n B since x is not an element of A

 

[kidmode01]

When proving these sorts of problems it is important to know what you need to show. What does it mean for two sets to be equal? It means that each set is a subset of each other. ie: Supposed A and B are two sets. If A=B then $$A \subseteq B$$ and $$B \subseteq A$$

It is also important to know what kind of proof we are dealing with. In this case it is an if and only if. So that means we have to prove both ways. First we prove If A n B = U then A = U and B = U. Second we prove the other way, If A = B and B = U then A n B = U.

So you started off proving the one way. Suppose A n B = U. You do not need to suppose A is a proper subset of U because you are given that by definition in your problem. So you said x is an element of B, and then x is not an element of A n B because x is not an element of A. It very well may be in A! We do not have enough information to conclude that if we pick an element in B, it can't be in A. Instead we should pick an element x inside A n B. Then x is an element of A and x is an element of B. Since A and B are subsets of U, x is an element of U...see where I'm going? We need to show A = U and B = U. That means we need to show A is a subset of U and U is a subset of A. Similarly for B. Well we already know that A is a subset of U by definition. But is U a subset of A? What information do we have? Start by picking an element out of U and showing that it is inside A using our assumptions. Then U will be a subset of A and thus A=U. A similar argument will be made for B.

Then we have to prove the other way. If A=U and B=U then A n B = U.

I hope this helps.

 

[mbcsantin]

When proving these sorts of problems it is important to know what you need to show. What does it mean for two sets to be equal? It means that each set is a subset of each other. ie: Supposed A and B are two sets. If A=B then $$A \subseteq B$$ and $$B \subseteq A$$

It is also important to know what kind of proof we are dealing with. In this case it is an if and only if. So that means we have to prove both ways. First we prove If A n B = U then A = U and B = U. Second we prove the other way, If A = B and B = U then A n B = U.

So you started off proving the one way. Suppose A n B = U. You do not need to suppose A is a proper subset of U because you are given that by definition in your problem. So you said x is an element of B, and then x is not an element of A n B because x is not an element of A. It very well may be in A! We do not have enough information to conclude that if we pick an element in B, it can't be in A. Instead we should pick an element x inside A n B. Then x is an element of A and x is an element of B. Since A and B are subsets of U, x is an element of U...see where I'm going? We need to show A = U and B = U. That means we need to show A is a subset of U and U is a subset of A. Similarly for B. Well we already know that A is a subset of U by definition. But is U a subset of A? What information do we have? Start by picking an element out of U and showing that it is inside A using our assumptions. Then U will be a subset of A and thus A=U. A similar argument will be made for B.

Then we have to prove the other way. If A=U and B=U then A n B = U.

I hope this helps.

thanks! it really helps!