Describe the steps in and/or define how to accomplish the following types of proofs:
a. That two sets are equal.
b. That two sets are disjoint.
c. A proof by contra-positive.
d. A proof by contradiction.
e. A proof by Mathematical Induction.
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a. Proof that two sets are equal:
I'll call the sets A and B. Here, we want to show that the elements of each set are also in the other set, and that there are no extras in either set. So what we need to do is to show that if something is in set A, it's also in set B, and that if something is in set B, it's also in set A. That is, we want to show that if x is in A, that implies that x is in B, and that if y is in B, that implies that y is in A.
b. Proof that two sets are disjoint:
Here we want to show that none of the elements of either set are in the other set (remember that this is what "disjoint" means: the sets have nothing in common). So similarly to above, we want to show that if x is in A, that implies that x is NOT in B, and that if y is in B, that implies that y is NOT in A.
c. Proof by contrapositive:
Now I'll let A be my set of assumptions (the given information) and B be the thing I want to prove. The proof is then supposed to show that A => B (A implies B, our assumptions imply the conclusion).
Recall that the contrapositive of P => Q (P ...
This solution describes how to compose two types of proofs about sets, as well as proofs by contrapositive, by contradiction, and by induction.