1. 1 CA 208 Logic Ex1 In your own words, define the following 1. Logic: 2. Valid reasoning/inference (2 equivalent definitions): 3. Propositions/statements:  List the 1. 4 binary 2. 1 unary connectives (use the special symbols)  Translate the following into plain English 1. P |= C 2. {A,B} |= A  B 3. {A  B} |= A 4. {A, A → B} |= B

2. 2 CA 208 Logic Ex1 Formalise the following arguments/inferences using proposional variables (P, Q, R,...) and the logical connectives. State the translation keys: 1. Kate is a student. If Kate is a student, then Kate is broke. |= Kate is broke. 2. Kate is a student. Kate is broke. |= Kate is a student and Kate is broke. 3. Kate is a student and Kate is broke. |= Kate is a student. 4. Kate is a student. |= Kate is a student. 5. Kate is taller than John. John is taller than Mike. If Kate is taller than John and John is taller than Mike, then Kate is taller than Mike. |= Kate is taller than Mike.  Intutively, are the inferences above logically valid (i.e. is the conclusion true in all situations where the premises are true)?  Is the following inference logically valid? 1. If Kate is a student, then Kate is broke. Kate is broke. (|= ??) Kate is a student.

3. 3 CA 208 Logic Ex1 Complete the following truth tables: PQ PQPQPQPQPQPQPQPQ 11 10 01 00 PQ PQPQ  P  Q  P   Q  (  P   Q) 11 10 01 00 P PP 1 0 P PP   P 1 0

4. 4 CA 208 Logic Ex2 What are  Tautologies  Contradictions  Contingencies Define logical equivalence  in terms of  Intuitively, what does it mean for two formulas to be logically equivalent?

5. 5 CA 208 Logic Ex2 Use the truth table method to show whether the following are tautologies, contingencies or contradictions  (P  (  P  Q))  Q   (  P   Q)  (P  Q)   (Q  Q) Use the truth table method and the definition of logical equivalence in terms of the biconditional (iff) to show that   (  P   Q)  (P  Q)

6. 6 CA 208 Logic Ex2 Use the Boolean equivalences to show (i.e.rewrite) that the following are logically equivalent:   (  Q   P)  (P  Q)  (P  (  Q   P))  (P   Q)