Sets

The idea of a set underlies almost every branch of modern mathematics, including relations, functions, geometry, sequences and probability. A set is a well-defined collection of objects, which means we can always decide whether a given object belongs to it. This chapter explains how to represent sets in roster form, by listing their elements, and in set-builder form, by stating the property shared by their elements. You will meet the empty set, finite and infinite sets, equal sets, subsets, proper subsets and the power set. A set with n elements has 2^n subsets. Intervals such as (a, b), [a, b], [a, b) and (a, b] are special subsets of the real numbers. The universal set contains all the sets under discussion, and Venn diagrams show the relationship between sets using circles inside a rectangle. You will learn the operations of union, intersection, difference and complement, together with their properties. These include the commutative, associative and distributive laws and De Morgan's laws, (A union B)' = A' intersection B' and (A intersection B)' = A' union B'.

Detailed Worksheet: Sets Section A - Definitions (10 marks) 1. Write the set {x : x is a prime number less than 10} in roster form. (2 marks) 2. Write the set {1, 4, 9, 16, 25} in set-builder form. (2 marks) 3. Write the power set of {1, 2} and state how many subsets it has. (2 marks) 4. Write the interval [-3, 5) in set-builder form. (2 marks) 5. Give one example each of an empty set and an infinite set. (2 marks) Section B - Calculations and Applications (15 marks) 6. If A = {1, 3, 5, 7, 9} and B = {2, 3, 5, 7}, find A union B, A intersection B, A - B and B - A. (3 marks) 7. If U = {1, 2, 3, 4, 5, 6, 7, 8, 9} and A = {1, 2, 3, 4}, find A' and verify that A union A' = U. (3 marks) 8. Are the sets {x : x is a letter of the word FOLLOW} and {x : x is a letter of the word WOLF} equal? Give reasons. (3 marks) 9. Distinguish between a subset and a proper subset with examples. How many proper subsets does a set with 3 elements have? (3 marks) 10. Which of the following pairs of sets are disjoint: (i) {1, 2, 3, 4} and {x : x is a natural number and 4 <= x <= 6} (ii) {a, e, i, o, u} and {c, d, e, f}? (3 marks) Section C - Diagrams (10 marks) 11. Draw Venn diagrams to show (i) A union B (ii) A intersection B (iii) A - B. (4 marks) 12. Draw a Venn diagram to show the complement A' of a set A in a universal set U. (3 marks) 13. Represent the intervals [2, 5) and (-1, 3] on the number line. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. If U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8} and B = {2, 3, 5, 7}, verify both of De Morgan's laws. (5 marks) 15. State and explain the laws of the algebra of sets: idempotent, identity, commutative, associative and distributive laws. (5 marks) 16. If A = {1, 2, 3, 4}, B = {3, 4, 5, 6} and C = {5, 6, 7, 8}, verify that A union (B intersection C) = (A union B) intersection (A union C). (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. A' denotes the complement of A. Write sets in roster form unless told otherwise.