Chapter 1.4, Problem 11E

Exercises throughout the text with this title ask you to examine “Proofs to Grade.” These are claims alleged to be true and supposed “proofs” of the claims. You should decide the merit of the claim and the validity of the proof, and then assign a grade of
A (correct), if the claim and proof are correct, even if the proof is not the simplest or the proof you would have given.
C (partially correct), if the claim is correct and the proof is largely correct. The proof may contain one or two incorrect statements or justifications, but the errors are easily correctable.
F (failure), if the claim is incorrect, or the main idea of the proof is incorrect, or there are too many errors.

You must justify assignments of grades other than A, and if the proof is incorrect, you must explain what is incorrect and why.
(a) Suppose a is an integer.
Claim. If a is odd, then $a^{\text{2}}+\text{1}$ is even.
“Proof.” Let a. Then, by squaring an odd we get an odd. An odd plus an odd is even. So $a^{\text{2}}+\text{1}$ is even.
(b) Suppose a, b, and c are integers.
Claim. If a divides b and a divides c, then a divides $b+c$ .
“Proof.” Suppose a divides b and a divides c. Then for some integer $q,b=aq$ , and for some integer $q,c=aq$ . Then $b+c=aq+aq=\text{2}aq=a(\text{2}q)$ , so a divides $b+c$ .
(c) Suppose x is a positive real number.
Claim. The sum of x and its reciprocal is greater than or equal to 2.
That is,
   $x+\frac{1}{x}\ge 2$ .
“Proof.” Multiplying by x, we get $x^{\text{2}}+\text{1}\ge \text{2}x$ . By algebra, $x^{\text{2}}-\text{2}x+\text{1}\ge 0$ . Thus, ${(x-\text{1})}^{\text{2}}\ge 0$ . Any real number squared is greater than or equal to zero, so $x+\frac{1}{x}\ge 2$ is true.
(d) Suppose m is an integer.
Claim. If $m^{\text{2}}$ is odd, then m is odd.
“Proof.” Assume m is odd. Then $m=\text{2}k+\text{1}$ for some integer k. Therefore, $m^{\text{2}}=(\text{2}k+\text{1})\text{2}=\text{4}{k}^{\text{2}}+\text{4}k+\text{1}=\text{2}(\text{2}{k}^{\text{2}}+\text{2}k)+\text{1}$ , which is odd. Therefore, if $m^{\text{2}}$ is odd, then m is odd.
(e) Suppose a is an integer.
Claim. $a^{\text{3}}+a^{\text{2}}$ is even.
“Proof.” $a^{\text{3}}+a^{\text{2}}=a^{\text{2}}(a+\text{1})$ , which is always an odd number times an even number. Therefore, $a^{\text{3}}+a^{\text{2}}$ is even.