# State whether the

(a) The set of all even natural numbers is a smallest element – False

The set of all even natural number consist of elements (2, 4, 6, 8,10,……) but the smallest element is 0 which is not included in the set of natural numbers and the smallest natural number is 1 which is also not included in set even numbers.

(b) Every non-empty subset of has the smallest element. – true

Every non empty subset of Z have the smallest element. It depends on the conditions attached to the non-empty subset such as the set of even natural numbers has 2 as its least element. The set of prime numbers bigger than 5 has 7 as its least element

(c ) Every integer can be identified with a rational number- true

Every integer is a rational number as each integer n can be written in the form .

For example 7 = 7/1 and thus 7 is a rational number.

(d) For each rational number, one can find the next rational number.- False

One cannot always find the next rational number for each rational number as there also lies irrational numbers in-between.

(e) There is the largest rational number.- False

The largest rational number cannot be determined as there are infinites numbers and each number can be expressed as a rational number.

(f) Every integer is either even or odd – true

We know n is even if n = 2k for some integer k and n is odd if n = 2k + 1 for some integer k.

Every integer can be expressed as 2k or 2k + 1 hence every integer is either odd or even.

(g) Between any two rational numbers, there is an integer. - false

Rational number can be any number 5/1, 2/1, which can be expressed in form of fraction and integers are the whole numbers which are not a fraction so there cannot lie a whole number between two rational number which are fractions.

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