7:51 1015 a brightspace.meredith.edu WA 6--Writing Proots using different methods of argumentation Written Assignment 6 Writing Proofs using different methods of argumentation. In Chapter 3, we have discussed several different types of proof structures that use different methods of argumentation. We want to be sure that you can communicate a well-written proof in each style of argumentation. Write a well-constructed proof of each of the following theorems. 1. Direct Proof: (Transitive property of congruence mod n) For all integers a, b, and c, if a = b (mod n) and b = c (mod n), then a = c (mod n). 2. Proof by Contrapositive: Let x and y denote positive real numbers. If x #y, then x + y > 4xy x + y 3. Proof by Contrapositive: Let a be a positive real number. If a is irrational, then a is irrational. 4. Biconditional Proof: Let a Z. Then, a is odd if and only if a can be written as the sum of two consecutive integers. KY 5. Existence Proof: There exists a sequence of 100 consecutive positive integers, none of which is prime. What type of existence proof is the proof of the theorem above? Explain your answer 13%