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Problem 2 Prove that for all integers n, it is the case that n is even if and only if 3n is even. This proof has two parts: you must show that n being even implies that 3n is even; then you must show that 3n being even implies that n is even. In your proof, please one of the two directions using direct proof. Then, prove the other direction using proof by contrapositive.

Problem 2 Prove that for all integers n, it is the case that n is even if and only if 3n is even. This proof has two parts: you must show that n being even implies that 3n is even; then you must show that 3n being even implies that n is even. In your proof, please one of the two directions using direct proof. Then, prove the other direction using proof by contrapositive.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Help me understand and solve these two problems about direct proofs and proof by contrapositive. Thank you 

For each of the following four statements: First, represent the statement using logical symbols, including quantifiers. Define any predicates you need. • For example, a symbolic representation of the first statement would be \m € Z : E(m) → E(7m + 4), where E(x) is the predicate "x is even." Then, write a careful proof of the statement, justifying each of your steps. You do not need to use logical symbols in your proofs. ● 1. If m is an even integer then 7m + 4 is an even integer. 2. If m is an even integer and n is an odd integer then m + n is an odd integer. 3. If m is an even integer and n is an odd integer then mn is an even integer. 4. If a, b, and care integers such that a divides b and b divides c, then a divides c. Notation note: For some of these statements, you may find yourself needing to write things like Vx € Z, Vy E Z. A common notational shortcut is to write Vx, y E Z to mean the same thing while saving some space.
Transcribed Image Text:For each of the following four statements: First, represent the statement using logical symbols, including quantifiers. Define any predicates you need. • For example, a symbolic representation of the first statement would be \m € Z : E(m) → E(7m + 4), where E(x) is the predicate "x is even." Then, write a careful proof of the statement, justifying each of your steps. You do not need to use logical symbols in your proofs. ● 1. If m is an even integer then 7m + 4 is an even integer. 2. If m is an even integer and n is an odd integer then m + n is an odd integer. 3. If m is an even integer and n is an odd integer then mn is an even integer. 4. If a, b, and care integers such that a divides b and b divides c, then a divides c. Notation note: For some of these statements, you may find yourself needing to write things like Vx € Z, Vy E Z. A common notational shortcut is to write Vx, y E Z to mean the same thing while saving some space.
Problem 2 Prove that for all integer n, it is the case that n is even if and only if 3n is even. This proof has two parts: you must show that ʼn being even implies that 3n is even; then you must show that 3n being even implies that n is even. In your proof, please one of the two directions using direct proof. Then, prove the other direction using proof by contrapositive.
Transcribed Image Text:Problem 2 Prove that for all integer n, it is the case that n is even if and only if 3n is even. This proof has two parts: you must show that ʼn being even implies that 3n is even; then you must show that 3n being even implies that n is even. In your proof, please one of the two directions using direct proof. Then, prove the other direction using proof by contrapositive.
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