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Consider the statement If two integers have different parities, then their sum is odd. Which of the following provides a correct, complete and logical proof of the above statement? Proof II Suppose a and are two integers with different parities. • Without loss of generality, assume that a is even and b is odd. • Ep. q € Z such that a = 2p and b = 2q+1. • The sum a+b=2p+2q+1 = 2l+1, l € Z. Therefore a + b is odd. Proof IV • Let a = 5 and b = 8. • a+b=5+8=13 which is odd. Let a = 2 and b=3. • a+b=2+3=5 which is odd. Therefore the sum is always odd. Proof III Suppose that a is even and b is odd. 3p, q E Z such that a = 2p and b = 2q+1. The sum a+b=2p+2q+1 = 2l+1, l € Z. Therefore the sum odd. Proof I Suppose the sum of two integers a and b is odd. The sum of two even numbers is even; the sum of two odd numbers is even; the sum of an even and an odd number is odd. Therefore one of a or b (but not both) has to be odd, and the other even. Hence a and b have different parities.

Consider the statement If two integers have different parities, then their sum is odd. Which of the following provides a correct, complete and logical proof of the above statement? Proof II Suppose a and are two integers with different parities. • Without loss of generality, assume that a is even and b is odd. • Ep. q € Z such that a = 2p and b = 2q+1. • The sum a+b=2p+2q+1 = 2l+1, l € Z. Therefore a + b is odd. Proof IV • Let a = 5 and b = 8. • a+b=5+8=13 which is odd. Let a = 2 and b=3. • a+b=2+3=5 which is odd. Therefore the sum is always odd. Proof III Suppose that a is even and b is odd. 3p, q E Z such that a = 2p and b = 2q+1. The sum a+b=2p+2q+1 = 2l+1, l € Z. Therefore the sum odd. Proof I Suppose the sum of two integers a and b is odd. The sum of two even numbers is even; the sum of two odd numbers is even; the sum of an even and an odd number is odd. Therefore one of a or b (but not both) has to be odd, and the other even. Hence a and b have different parities.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Consider the statement If two integers have different parities, then their sum is odd. Which of the following provides a correct, complete and logical proof of the above statement? Proof II Suppose a and are two integers with different parities. • Without loss of generality, assume that a is even and b is odd. • Ep. q € Z such that a = 2p and b = 2q+1. • The sum a+b=2p+2q+1 = 2l+1, l € Z. Therefore a + b is odd. Proof IV • Let a = 5 and b = 8. • a+b=5+8=13 which is odd. Let a = 2 and b=3. • a+b=2+3=5 which is odd. Therefore the sum is always odd. Proof III Suppose that a is even and b is odd. 3p, q E Z such that a = 2p and b = 2q+1. The sum a+b=2p+2q+1 = 2l+1, l € Z. Therefore the sum odd. Proof I Suppose the sum of two integers a and b is odd. The sum of two even numbers is even; the sum of two odd numbers is even; the sum of an even and an odd number is odd. Therefore one of a or b (but not both) has to be odd, and the other even. Hence a and b have different parities.
Transcribed Image Text:Consider the statement If two integers have different parities, then their sum is odd. Which of the following provides a correct, complete and logical proof of the above statement? Proof II Suppose a and are two integers with different parities. • Without loss of generality, assume that a is even and b is odd. • Ep. q € Z such that a = 2p and b = 2q+1. • The sum a+b=2p+2q+1 = 2l+1, l € Z. Therefore a + b is odd. Proof IV • Let a = 5 and b = 8. • a+b=5+8=13 which is odd. Let a = 2 and b=3. • a+b=2+3=5 which is odd. Therefore the sum is always odd. Proof III Suppose that a is even and b is odd. 3p, q E Z such that a = 2p and b = 2q+1. The sum a+b=2p+2q+1 = 2l+1, l € Z. Therefore the sum odd. Proof I Suppose the sum of two integers a and b is odd. The sum of two even numbers is even; the sum of two odd numbers is even; the sum of an even and an odd number is odd. Therefore one of a or b (but not both) has to be odd, and the other even. Hence a and b have different parities.
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