Rewrite each of the following sentences using mathematical notation (∀, ∃, ...) and state whether the sentence is true or false with a brief reason.

(a) For every integer there is a smaller real number.

(b) There exists a real number that is smaller than every integer.

(a) For each integer there is a smaller real number, it is equivalent to the expression: ∀x∈Z, ∃y∈R, so that, real number y is less than the integer number x. (Z represents the set of integers numbers and R represents the set of real numbers)

(b) There is a real number that is smaller than each integer, it is equivalent to the expression: ∃y∈R, such that, ∀x∈Z, that number y has to be less than integer number x. (Z represents the set of integers numbers and R represents the set of real numbers)