Problem 1 (Relations) [25 marks]
1. Show that the relation
R = f(x; y)j (x y) is an even integerg
is an equivalence relation on the set R of real numbers.
2. Show that the relation
R = f((x1; y1); (x2; y2)) (x1 <x2) or ((x1 = x2) and (y1 y2))g
is a total ordering relation on the set R R.
Problem 2 (Basic probability calculations) [25 marks] In a roulette, a wheel with 38 numbers is spun. Of these, 18 are red, and 18 are black. The other two numbers, which are neither black nor red, are 0 and 00. The probability that when the wheel is spun it lands on any particular number is 1/38.
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What is the probability that the wheel lands on a red number?
What is the probability that the wheel lands on a black number twice in a row?
What is the probability that the wheel lands on 0 or 00?
What is the probability that in ve spins the wheel never lands on either 0 or 00?
Provide detailed justi cations of your answers.
Problem 3 (Bayes theorem) [25 marks] Suppose that 8% of all bicy-cle racers use steroids, that a bicyclist who uses steroids tests positive for steroids 96% of the time, and that a bicyclist who does not use steroids tests positive for steroids 9% of the time (this is a \false positive" test result).
What is the probability that a randomly selected bicylist who tests positive for steroids actually uses steroids?
What is the probability that a randomly selected bicyclist who tests negative for steroids did not use steroids?
Problem 4 (Graphs) [25 marks] For each of the following two graphs, determine whether or not it has an Euler circuit. Justify your answers. If the graph has an Euler circuit, use the algorithm described in class to nd it, including drawings of intermediate subgraphs.
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