# Fill in the blank

(a) The number 0 is not in the set of natural numbers.

0 is added to set of natural number (1,2,3,4,…..) and it becomes set of whole numbers (0,1,2,3,4,….)

(b) The least number in the set of all the whole numbers is 0.

0 is the smallest number which when added to the set of natural number make it a set of whole number.

(c) The least number in the set of all even natural numbers is 2.

The set of even natural number starts from (2,4,6,….) so the least number in the set of all even natural numbers is number 2 .

(d) The successor of 8 in the set of all natural numbers is 9.

The number succeeding number 8 is number 9 in the set of all natural numbers as the set of natural number consist of term (1,2,3,4,5,6,7,8,9,10,11,12,…..)

(e) The sum of two odd integers is always an even number

for e.g;- 3 + 5 = 8 , 5 + 7 = 12

Let m and n be odd integers. By definition of odd we have that m = 2a + 1 and

n = 2b + 1. Consider the sum m + n = (2a + 1) + (2b + 1) = 2ab + 2b + 2 = 2k, where

k = a + b + 1 is an integer. Therefore by definition of even we have shown that

m + n is even.

(f) The product of two odd integers is odd.

Let a and b be two odd integers.

By definition of odd we know that a = 2m + 1 and b = 2n + 1.

Now, the product ab = (2m + 1)(2n + 1)

= 4mn + 2m + 2n + 1

= 2(2mn + m + n) + 1 = 2k + 1, where k = (2mn + m + n ) is an integer. Therefore by

Definition of odd number, the product of two odd integers is also odd.

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