 # Q1 For each of the functions below. state whether f is one-to-one, ...

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Q1 For each of the functions below. state whether f is one-to-one, and whether f is onto. You do not need to show your working. (a) f: z-z. where f(2)=13: = (b) f: 0,00) - [0.oc), where f(z) = 22 (c) f: 11, 1], where f(z) = sinz: (d) f: - N. where f( (a,b)) = 2 -4° Q2 We saw in lectures how to write integers in base n. We can also write arbitrary real numbers in base n. using a "decimal point": (dgd. = -din+dd For example, (100.101)2 = 2² (a) Write the fraction 11/8 in base 2. using the form described above. (b) Write the fraction 1/4 in base 3. using the form described above. You should justify why your answers are correct. Q3 Let A. B and C be any subsets of some universal set x. Determine whether each of the following statements is true or false. If true, give a proof. If false, give a counterexample. (a) AU(B-A)= (AUB)-(AnB) - = - (b) Ae = - - (c) 1x(BC)=(4x B)-(Ax c) = - (d) P(A - B) = P(A) - P(B) Q4 Describe the following sets, and prove your answers correct. Your proofs should use the "element method" (which we called the "bare hands" method in lecture 18). 00 (a) UR.49 (b) 1 ] (c) Q5 Recall the Fibonacci sequence defined by Fo = 0, F1 = 1 and F. = Using induction, prove that = all +Q6 Let x and Y be any non-empty sets, and let ! :X - Y be any function from X to Y. (a) Show that for all A In x. (b) Give an example of a function f and a set A for which /-(f(()) (c) Show that, if f is injective, then =f(((()()) for all Agx. Once again, your proofs for (a) and (c) should use the "element method" (which we called the "bare hands" method in lecture 18)

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