Problem 122 Which of the following are true Justify your ans

Problem 12.2. Which of the following are true? Justify your answer.

(1) A rational number is an element of Z × (Z {0}).

(2) A rational number is an infinite set.

(3) Every infinite subset of Z × (Z {0}) is a rational number.

Solution

Solution:-

1.A rational number can be formally defined as the equivalence classes of the quotient set (Z x( Z-{0} ) ) / ~, where the cartesian product of ZX ( Z- {0}) is the set of all ordered pairs ( m, n) where m and n are integers.

Rational number Q refers to the fact that it is a ratio of two integers.

Hence statement 1 is false.

2) A rational number is in the form of p/q ,where p and q are some integers. A rational number is not an infinite set .But, set rational numbers between two integers is an infinite set.

Hence statement 2 is false.

3)Every infinite subset of ZX(Z -{0}) is a rational number, as every infinite subset consists of equivalence class of the quotient set, which is rational number.

Hence statement 3 is true.