Relations and Functions

Relations and Functions builds on the ideas of Class 11 and gives precise language for describing how elements of sets are connected, a foundation for the whole of higher mathematics. In this chapter you will study types of relations on a set: the empty and universal relations, and reflexive, symmetric and transitive relations. A relation that is all three is an equivalence relation, and you will see how it divides a set into disjoint equivalence classes, for example the classes of integers that leave the same remainder when divided by 4. You will then study types of functions. A function is one-one (injective) when different inputs always give different outputs, and onto (surjective) when every element of the codomain is the image of some element of the domain; a function that is both is a bijection. You will test functions such as f(x) = 3 - 4x, x^2 and x^3 on different domains, use graphs and algebra to decide their type, and count relations and functions between finite sets.

Detailed Worksheet: Relations and Functions Section A - Definitions (10 marks) 1. Define reflexive, symmetric and transitive relations on a set A. (2 marks) 2. What is an equivalence relation? What is an equivalence class? (2 marks) 3. Define a one-one function and an onto function. (2 marks) 4. Give an example of a relation on {1, 2, 3} that is symmetric but neither reflexive nor transitive. (2 marks) 5. Show that the empty relation on a non-empty set is symmetric and transitive but not reflexive. (2 marks) Section B - Calculations and Applications (15 marks) 6. Check whether the relation R = {(x, y): y = x + 1} on the set A = {1, 2, 3, 4, 5, 6} is reflexive, symmetric or transitive. (3 marks) 7. Show that the relation R = {(x, y): abs(x - y) is a multiple of 4} on A = {0, 1, 2, ..., 12} is an equivalence relation. Find the set of all elements related to 1. (3 marks) 8. Let A = {1, 2, 3} and B = {p, q}. Find the number of functions from A to B, the number of one-one functions and the number of onto functions. (3 marks) 9. Show that f: R -> R defined by f(x) = 3 - 4x is one-one and onto. (3 marks) 10. Check whether (i) f: N -> N, f(x) = x^2 and (ii) g: Z -> Z, g(x) = x^2 are one-one and onto. (3 marks) Section C - Diagrams (10 marks) 11. Draw arrow diagrams of four functions from a set of three elements to a set of three elements that are respectively one-one and onto, one-one but not onto (codomain of four elements), onto but not one-one, and neither. (4 marks) 12. Draw the graph of f(x) = x^2 on R and use the horizontal line test to explain why it is not one-one. (3 marks) 13. Draw a diagram showing how the equivalence relation of question 7 partitions the set {0, 1, ..., 12} into equivalence classes. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. Let f: N -> N be defined by f(n) = (n + 1)/2 if n is odd and f(n) = n/2 if n is even. Determine whether f is one-one, onto or bijective, with reasons. (5 marks) 15. Show that the relation R on the set of all triangles, defined by T1 R T2 if T1 is similar to T2, is an equivalence relation. For right triangles T1, T2 and T3 with sides 3, 4, 5; 5, 12, 13 and 6, 8, 10, state which are related. (5 marks) 16. Find the number of equivalence relations on {1, 2, 3} containing (1, 2) and (2, 1), listing them. Also give a relation on this set that is reflexive and symmetric but not transitive. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Justify every property with a general proof or a counter-example.