Simplify:

(i) 100(100 – 3) – (100 × 100 – 3);

(ii) (20–(2011–201) + (2011–(201–20))

Fill in the blanks:

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

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

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

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

(e) The sum of two odd integers is _____________.

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

State whether the following statements are true or false:

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

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

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

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

(e) There is the largest rational number.

(f) Every integer is either even or odd.

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

Suppose m is an integer such m ≠ –1 and m ≠ –2. Which is larger or ? State your reasons.

Define an operation * on the set of all rational numbers □ as follows:

r*s = r + s –(r × s),

for any two rational numbers r, s. Answer the following with justification:

(i) Is ℚ closed under the operation *?

(ii) Is * an associative operation of ℚ ?

(iii) Is * a commutative operation of ℚ?

(iv) What is a * 1 for any a in ℚ?

(v) Find two integers a ≠ 0 and b ≠ 0 such that a * b = 0.