Question 1: If is the statement is even, then what is ?

Answer:

Given

Question 2: If is the statement is divisible by , prove that is true but is not true.

Answer:

Given is divisible by

which is divisible by is true.

which is not divisible by is not true.

Question 3: If is the statement , and, if is true; prove that is true.

Answer:

Given

Given is true

Since

is true

Question 4: If is the statement is even, and. if is true, then is true.

Answer:

Given is even

Given is true is even

where

where

is even is true

Question 5: Give an example of a statement such that it is true for all .

Answer:

is true for all

Question 6: If is the statement is prime, prove that and are true. Prove also that is not true.

Answer:

Given is prime

is a prime number. Therefore is true.

is a prime number. Therefore is true.

is a prime number. Therefore is true.

is not a prime number. Therefore is not true.

Question 7: Give an example of a statement which is true for all but and are not true. Justify your answer.

Answer:

Let

. Therefore is not true.

. Therefore is not true.

. Therefore is not true.

. Therefore is true.

. Therefore is true.