Introduction to Proofs Types

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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...plies Q) is ~Q => ~P (not Q implies not P) and that these are equivalent statements. So if we have a set of assumptions A that we want to use to prove a statement B, equivalently we can show that ~B => ~A. So we assume the opposite of the thing we wanted to prove, and show that that implies the opposite of our assumptions.

d. Proof by contradiction:
This is similar to a proof by contrapositive, but it's not exactly the same. In a proof by contradiction, we take our assumptions A as normal, but we assume that the thing we want to prove, B, is false. Then we take this together (A and ~B) and show that somehow there's a contradiction; that our assumptions and the opposite (negation) of B cause something to be true and false at the same time, that sort of thing. An example of a contradiction is if we could show that A and ~B meant that both x<y and x>y. What this shows is that since we assume that our assumptions are true (they wouldn't be very good assumptions if not), the problem lies not with A but with ~B... ~B must have been false, which means that as long ...