1) Find two square numbers, each less than 100, which are also cube numbers.

1 and 64 are square numbers as well as cube numbers.

1 = 1 × 1

  = 1 × 1 ×1

64 = 8 × 8

    = 4 × 4 × 4

2) List these sets:

a. The set consisting of the first five odd numbers.

b. The set whose objects are the last four letters of the English alphabet.

c. The set of even numbers between 1 and 7.

a. The first five odd numbers are 1, 3, 5, 7, 9

∴ required answer = {1, 3, 5, 7, 9}

b. Last four letters of English alphabets are w, x, y, z.

∴ required answer = {w, x, y, z}

c. Even numbers between 1 and 7 are 2, 4, 6

∴ required answer = {2, 4, 6}

3) Describe these sets.

a. A = {2, 3, 5, 7}

b. B = {5, 10, 15, 20, 25}

a. 2, 3, 5, 7 are first four prime numbers.

∴ A = The set of the first four prime numbers.

b. 5, 10, 15, 20, 25 are the first five multiples of 5.

∴ B = The set of the first five multiples of 5.

4) A = {1, 2, 3, 4, 5, 6}, B = {2, 4, 6, 8, 10}, C = {3, 5, 7, 9, 11}. List the sets given by

a. {x : xA, x > 3}

b. {x : xB, x ≤ 6}

c. {x : xC, 5 < x < 12}

a. {x : x  A, x > 3} = {4, 5, 6}

b. {x : x  B, x ≤ 6} = {2, 4, 6}

c. {x : x  C, 5 < x < 12} = {7, 9, 11}

5) Express in set builder notation  of N :

a. Greater than 5

b. Less than 10

c. Between 3 and 11

d. Which are prime

a. Natural numbers greater than 5 are 6, 7, 8, 9, ...........

∴ Required answer = {6, 7, 8, 9.....}

b. Natural numbers less than 10 are 1, 2, 3, 4, 5, 6, 7, 8, 9

∴ Required answer = {1, 2, 3, 4, 5, 6, 7, 8, 9}

c. Natural numbers between 3 and 11 are 4, 5, 6, 7, 8, 9, 10

∴ Required answer = {4, 5, 6, 7, 8, 9, 10}

d. The prime numbers are 2, 3, 5, 7, .........

∴ Required answer = {2, 3, 5, 7, ........}

6) Find a universal set for each of the following set.

a. The set of people in your class that have long hair.

b. The set of vowels

c. {2, 4, 6, 8}

The set of elements from which to select to form subsets is called the Universal set and is denoted by U.

a. U = The set of people in your class.

b. U = The set of English alphabets.

c. U = The set of even natural numbers.

7) U = {1, 2, 3, 4, 5, 6}, A = {1, 2, 3, 4}, B = {3, 4, 5}, C = {5}

List the sets A∩B, B∩C, A∩C and U∩B.

A ∩ B = Elements that are found in both A and B = {3, 4}

B ∩ C = Elements that are found in both B and C = {5}

A ∩ C = Elements that are found in both A and C = Φ

U ∩ B = Elements that are found in both U and B = {3, 4, 5}

8) Write down the union of the following sets.

a. A = {a, b, c} and B = {d, e, f}

b. A = {x} and B = {y}

AUB = Element in A or B or both.

       = {a, b, c, d, e, f}

b. AUB = {x, y}

9) U = {p, q, r, s, t, u} and Y = {p, r, u}. List the members of Y'.

Y' = {x : x  U and x ∉ Y}

    = {q, s, t}

10) W = {whole numbers} and subset T = {even numbers}. Describe in words the complement of T with respect to N.

Complement of T = Set of elements in W not in T.

                          = Set of odd numbers and Zero.