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.