State the converse, inverse and contrapositive of each of the following statements:

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]

1. State the converse, inverse and contrapositive of each of the following statements:
2. I will buy a house if I can save more than $1000 per month.
3. .
4. Determine whether the following proposition is a tautology:

.

Q−2: [4+4 marks]

1. Find the set of solutions of each of the linear congruence:
2. .
3. .
4. Suppose , where
5. If , find .
6. If , find .

• If , find

1. Is one-to-one? Explain.

Q−3: [4+4 marks]

1. 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”.

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

Q­−4: [4+4 marks]

1. 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