Answer to Question #211838 in Discrete Mathematics for Jaguar

Question

Question 24

Consider the statement 

If n is a multiple of 3, then 2n + 2 is not a multiple of 3.

The converse of the given statement is:

 If n is not a multiple of 3, then 2n + 2 is a multiple of 3.

1. True

2. False

Question 25

Consider the following statement, for all x  Z:

 If x + 1 is even, then 3x2

- 4 is odd. 

The correct way to start a direct proof to determine if the statement is true is as follows:

 Assume x is even, then x = 2k for some k  Z,

 then 3x2 – 4

 ie 3(2k)2

- 4

 ie ………..

1. True

2. False

Answer 1

Question 24

Consider the statement "If $n$ is a multiple of 3, then $2n + 2$ is not a multiple of 3". The converse of a statement is formed by switching the hypothesis and the conclusion. Hence, the converse of the given statement is "If $2+2n$ is not a multiple of 3, then $n$ is a multiple of 3".

Answer: 2. False

Question 25

Consider the following statement, "for all $x \in\Z$ : If $x + 1$ is even, then $3x^2- 4$ is odd. 

The correct way to start a direct proof to determine if the statement is true is as follows:

 Assume $x+1$ is even, then $x +1= 2k$ for some $k \in \Z$,  then $3x^2 – 4=3(2k-1)^2 – 4$ ...

Answer: 2. False