EXERCISE 1.8.1: Proving statements about odd and even integers with direct proofs. Each statement below involves odd and even integers. An odd integer is an integer that can be expressed as 2k+1, where k is an integer. An even integer is an integer that can be expressed as 2k, where k is an integer. Prove each of the following statements using a direct proof. (a) The sum of an odd and an even integer is odd. (b) The sum of two odd integers is an even integer. (c) The square of an odd integer is an odd integer.
EXERCISE 1.8.1: Proving statements about odd and even integers with direct proofs. Each statement below involves odd and even integers. An odd integer is an integer that can be expressed as 2k+1, where k is an integer. An even integer is an integer that can be expressed as 2k, where k is an integer. Prove each of the following statements using a direct proof. (a) The sum of an odd and an even integer is odd. (b) The sum of two odd integers is an even integer. (c) The square of an odd integer is an odd integer.
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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Transcribed Image Text:Additional exercises
* EXERCISE
1.8.1: Proving statements about odd and even integers with direct proofs.
Each statement below involves odd and even integers. An odd integer is an integer that can be expressed as 2k + 1, where k is an
integer. An even integer is an integer that can be expressed as 2k, where k is an integer.
Prove each of the following statements using a direct proof.
(a) The sum of an odd and an even integer is odd.
(b) The sum of two odd integers is an even integer.
(c) The square of an odd integer is an odd integer.
(d) The product of two odd integers is an odd integer.
(e) If x is an even integer and y is an odd integer, then x2 + y? is odd.
(f) If x is an even integer and y is an odd integer, then 3x + 2y is even.
(g) If x is an even integer and y is an odd integer, then 2x + 3y is odd.
(h) The negative of an odd integer is also odd.
(1) If x is an even integer then (-1)* = 1
1) If x is an odd integer then (-1)* = -1
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Transcribed Image Text:EXERCISE
1.6.2: Truth values for quantified statements about integers.
In this problem, the domain is the set of all integers. Which statements are true? If an existential statement is true, give an example. If
universal statement is false, give a counterexample.
(a) 3x (x + x = 1)
(b) ax (x +2 = 1)
(c) vx (x2 - x + 1)
(d) vx (x2 - x+ 0)
(e) vx (x? > 0)
(f) ax (x2 > 0)
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