Prove the following using direct proof.a. If 5? − 3 = ? + 5, then ? = 2.b. If ? and ? are both odd, then ? + ? is even.c. The sum of two odd integers is even.2. Prove the following using proof by contrapositive.a. If ? and ? are even integers, then ? + ? is an even integer.b. If ? and ? are odd integers, then ?? is an odd integer.c. If ? is an odd integer, then ?3 + ? is even.3. Prove the following using proof by contradiction.a. If ?2is an even integer, then ? is an even integer.b. If ? and ? are odd integers, then ? + ? is an even integer.c. Prove that √2 is irrational. 1. Prove the following using direct proof.a. If 5x – 3 = x + 5, then x = 2.b. If x and y are both odd, then x + y is even.c. The sum of two odd integers is even.2. Prove the following using proof by contrapositive.a. If m and n are even integers, then m + n is an even integer.b. If x and y are odd integers, then xy is an odd integer.c. If n is an odd integer, then n3 +n is even.3. Prove the following using proof by contradiction.a. If n? is an even integer, then n is an even integer.b. If n and m are odd integers, then n +m is an even integer.c. Prove that 2 is irrational.

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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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Prove the following using direct proof.
a. If 5? − 3 = ? + 5, then ? = 2.
b. If ? and ? are both odd, then ? + ? is even.
c. The sum of two odd integers is even.
2. Prove the following using proof by contrapositive.
a. If ? and ? are even integers, then ? + ? is an even integer.
b. If ? and ? are odd integers, then ?? is an odd integer.
c. If ? is an odd integer, then ?
3 + ? is even.
3. Prove the following using proof by contradiction.
a. If ?
2
is an even integer, then ? is an even integer.
b. If ? and ? are odd integers, then ? + ? is an even integer.
c. Prove that √2 is irrational.