I got help with one proof already, I am just lost when it comes to proofs....here is another on...
Problem:
Prove that the empty set is unique. That is, suppose that A and B are empty sets and prove that A=B
If we are working in the Zermelo-Frankael Set theory model, we will use the axiom of extensionality.
"Two sets are equal if and only if for any element of A if and only if is an element of B".
Since A has no elements the statement,
"If x in A then x in B" is true.
Similarly,
"If x in B then x in A" is true.
These statements are true because the hypothesis of the conditional is false. And a false impling a true of a false impling a true is always true. Thus these statements are both true.
Hello, luckyc1423!
Here's a rather primitive approach . . .Prove that the empty set is unique.
That is, suppose that and are empty sets. .Prove that .
Let and
When are two sets and unequal?
They are unequal if there is an element in which is not in
. . . . . . . . . . .or if there is an element in which is not in .
(1) Is there an element in which is not in ?
. . . Since , the answer is No.
(2) Is there an element in whichis ot in ?
. . . Since , the answer is No.
Therefore: .