Question

Consider the following statement. ∀a ∈ Z, (a − 1) a is not an integer. (a) Select the correct negation for this statement. For every integer a, a − 1 a is an integer. There is an integer a such that a − 1 a is not an integer. For every integer a, a − 1 a is not an integer. There is an integer a such that a − 1 a is an integer. (b) Is the given statement true or false? If the statement is true, enter TRUE; if the statement is false, enter a value of a that could be used as part of a counterexample that justifies its falseness.

Answer 1

a) There is an integer a such that
`(a-1)/(a)` is an integer

b) False

Explanation:

Statement : ∀a ∈ Z, (a − 1) a is not an integer

A) The correct negation will be :

There is an integer a such that
`(a-1)/(a)` is an integer

B) The Given statement is FALSE because

when we assume the value of a = 1

( a - 1 ) / a = (1 - 1 ) / 1 = 0 ; which is an integer