A Discrete Maths Test

Discrete Mathematics is easily the most intriguing subject I've ever taken in my life. After studying this subject for 6 weeks, I think I just might have a little idea of what Discrete Maths is all about. Let me try to put it into perspective for you.

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The main idea is simple - To make a simple concept as complicated as possible, to the point where no ordinary people can understand it.

If it sounds too abstract, let me give you the simplest example.

Primary School: 1 + 1 = ? [1 Mark]

Secondary School: 1 + 1 = X. Find the value of X. [1 Mark]

That's what you've been getting all your life. Now, let's take a look at a sample Discrete Maths paper.

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Discrete Mathematics Sample Examination Paper (2008/2009)

Section A

Question 1: The set X is related to Y, denoted by XRY, where the universe of discourse of X is the set of all positive integers smaller than 2, and the domain of Y is the set of all positive odd integers smaller than 3, defined by the binary relation as follows: For All x in X and for all y in Y, XRY => x + y = 2. [50 Marks]

a) Write down the power set of X and Y. [2]

b) Is R reflexive? Is R transitive? Is R symmetric? Is R anti-symmetric? Justify your answer for each case. [8]

c) For each of the above case, use logical reasoning to prove whether XRY is true for all cases of x and all cases of y. If not, show a counterexample by proving there exists a x or y where R is not true. [5]

d) Least out the greatest, least, maximal and minimal elements with respect to R. [4]

e) Justify if R is a total order. [2]

f) Explain whether XRY is a Tautology and plot the truth table for XRY. [4]

g) Highlight all the critical rows of the truth table. [1]

h) Draw the Venn Diagram of the relation [2]

i) Draw the Arrow Diagram of the relation [2]

Now, expand X and Y to the set of all Z+ < 100, or the set of all positive integers less than 100.

j) Is the relation One to One? Is the relation Onto? Justify your answer and write down the inverse function if it exists. [6]

k) What is the last number of x required for all values of y? Prove and justify your answer [4]

l) Guess a general formula for the above relation, and prove it by mathematical induction. [10]

Section B

Now, supposed a random x is chosen out of X, where X is just a sample of a larger population, K, and K is the set of all positive integers smaller than 1000.

m) Find the probability, denoted by P(X = x), that x is equal to 1.

n) Find the mean and variance of the population of X.

o) Identify the probability distribution associated with X.

p) Use unbiased point estimate to infer the population mean.

q) Use interval estimate toe infer the population mean with a 95% confidence level.

r) If someone claims that the mean of x is 500, test at a significance level of 5% whether it is possible to reject his claim.

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z) You have finally arrived at the last part of this one and only question in this test. Give yourself a round of applause and congratulations for sitting through this ****. [1]

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