# 2 Functions a. A function f : A-B is called injective or one-to-one if whenever f(x)-f(y) for some x, y

###### Question: 2 Functions a. A function f : A-B is called injective or one-to-one if whenever f(x)-f(y) for some x, y E A then x = y. That is Vz, y A f(x) = f(y) → x = y. Which of the following functions are injective? In each case explain why or why not i. f:Z-Z given by f() 3r +7 (1 mark ii. f which maps a QUT student number to the last name of the student with that student number 1 mark b. Suppose that we have some finite set S and we have put thern into a list, 31, s2, . . . , sn (where n = ISI) We will argue that the list produces a topological ordering on S. To do this we first define f:sZ where f(s1)-1, f(s2)2 and in general f(sj)-j i. Argue that f is injective 1 mark ii. Define a relation R on S by aRb whenever f(a) S f(b). Argue that R is a total ordering on R by showing that R is reflexive, anti-symmetric, transitive, and has the total ordering property: Vz, y E S rRyVyRr Vr = y. (Hint: you will need to use the fact that f is injective to show that R is anti-symmetric.) (2 marks)

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