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

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