Can anyone help me with these proofs:
1. Prove that for any set A and B, the
where P(X) = power set of X.
2. Prove that for every real number x, there is a real number y such that
3. Suppose F and G are nonempty family of sets and every element of F is disjoint from some element of G. Prove that the union of all sets in F and the intersection of all sets in G are disjoint.
THANKS! MY MATH 109 GRADE DEPENDS ON THIS!
Suppose F and G are nonempty family of sets and every element of F is disjoint from some element of G. Prove that the union of all sets in F and the intersection of all sets in G are disjoint.
Let the families be respectively. We have that there is some element such that, . (1)
Suppose now the sentence "the union of all sets in and the intersection of all sets in are disjoint" is false. This means "the union of all sets in and the intersection of all sets in have a common element". Call this . Certainly, and (the intersection of this family). Now since , we must have that intersects all elements of the family: . This and (1) contradict each other.
Double inclusion: We show and .
For the first inclusion, let . We will prove .
We have , so and . Then and , which gives . qed
The other inclusion is up to you
2. Prove that for every real number x, there is a real number y such that
Double implication. We must show and .
For the first one. Suppose by contradiction and . Then , a contradiction.
For the second one. Suppose . Then choose for what Hacker has proposed. The proof is now complete.