Question

Consider the proposition " There exists an integer n such that for all integers m, m

a) Write the negation of the proposition (in English).

b) Determine if the proposition is true or if the negation is true. Give a reason for your choice.

Answer 1

a) “There is no integer n such that for all integers m, m<n”

b) The proposition is false and its negation true.

Explanation:

a)

The negation of the proposition is

“There is no integer n such that for all integers m, m<n”

b)

It is easy to prove that the proposition is false.

If there existed such an integer n , then since n+1 is an integer, n+1 < n, which is a contradiction.

So such an integer n cannot exist.

Since the proposition is false, its negation is true.