The TMA covers only chapters 1, 2, 4, 6 and 7 and consists of eight questions for a total of 40 marks. Please solve each question in the space provided. You should give the details of your solutions and not just the final results.

Q−1: [3+3 marks]

- State the converse, inverse and contrapositive of each of the following statements:
- I will buy a house if I can save more than $1000 per month.
- .
- Determine whether the following proposition is a tautology:

.

Q−2: [4+4 marks]

- Find the set of solutions of each of the linear congruence:
- .
- .
- Suppose , where
- If , find .
- If , find .

- If , find

- Is one-to-one? Explain.

Q−3: [4+4 marks]

- Find the smallest positive integer in the encryption function

,

such that the function encodes the letter “H” by “C”. Use the function to encrypt the message “NEED HELP”.

- Prove that: If is an integer that is not a multiple of 3, then .

Q−4: [4+4 marks]

- Let be the relation on the set defined by if and only if . Find the matrix representing .

Suppose that the relation is defined on the set where means . Show that is an equiva

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