Establish whether each of the following propositions is a tautology, contradiction or contingency:

Q−1: [4+2 marks]

1. Establish whether each of the following propositions is a tautology, contradiction or contingency:
2. , by using truth tables.
3. , by using laws of logic.
4. Let be the statement ”  is happy” and  be the statement ”  is under 20″, where the universe of discourse for  is the set of AOU students.  Express each of the following English statement in terms of quantifiers and logical connectives:
5. There is an AOU student who is happy and over 20.
6. Not all AOU students are either unhappy or under 20.

Q−2: [4+4 marks]

1. Define , where are any two sets. Show that .
2. Prove or disprove:
3. .
4. For all positive real numbers and , .

Q−3: [4+4 marks]

1. For all integers , determine whether each of the following is TRUE or FALSE, Justify your answer:
   1. If and , then .
   2. If and  then .

• If and , then .

1. If and  then .

1. Suppose that the encryption function

mod 26,  and ,

encrypt the letter “F” by “A”. Use the function  to decrypt the message “ZSVXO”.

Q­−4: [4+4 marks]

a)  Let  and  be relations on the set  represented by the matrices and .

1. List the elements of the set that represents the relation .
2. Show that is not transitive and find its transitive closure.
3. Find the equivalence relation  on  with partitions , and sketch its diagraph.