Answer to Question #238759 in Discrete Mathematics for Alina

Question

For the following, find if there are any errors in the methods of proof given below. List out these errors and write how you would prove/disprove the statements given below. (a) Statement: If n is an integer and n^2 is divisible by 4, then n is divisible by 4. Proof: Consider the number 144, which is a perfect square divisible by 4 ( since 4 × 36 = 144). Now, considering that √ 144 = 12 so n=12. Since 12 is also divisible by 4 (4 × 3 = 12), the statement holds true. Hence, Proved! (b) Statement: Let p and q be integers and r = pq + p + q, then r is even if and only if p and q are both even. Proof: Since p and q are even we can write them as p = 2k1 and q = 2k2. This means - r = 2k1 · 2k2 + 2k1 + 2k2, r = 2(2 · k1 · k2 + k1 + k2), r = 2(k3) Meaning r is an even number. Therefore, the statement above is true.

Answer 1

$a)$

$\text{An error has been made.}$

$\text{A general statement is derived from a particular statement. }$

$if \ n^2\vdots4\ then\ n\vdots4$

$\text{Counter-example:}$

$6^2=36\vdots4 - true; 6\vdots4-false$

$\text{Statement If n is an integer and } n^2 \text{ is divisible by 4, then n is divisible by 4 is false}$

$b)$

$\text{ Statement: Let p and q be integers and r = pq + p + q, then r is even if and only}$

$\text{if p and q are both even is true.}$

$\text{But for completeness of the proof, it is necessary to prove that}$

$\text{If one or both of the numbers p, q are odd then r is odd.}$

$\text{Let one of the numbers p, q be odd then:}$

$pq -\text {is even number}$

$p+q \text { is odd number}$

$pq +p+q \text { is odd number}$

$r \text { is odd number}$

$\text{Let both numbers p and q be odd then:}$

$pq -\text {is odd number}$

$p+q \text { is even number}$

$pq +p+q \text { is odd number}$

$r \text { is odd number}$