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CA 208 Logic Ex1

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

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CA 208 Logic Ex1

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

  2. CA 208 Logic Ex1 • Formalise the following arguments/inferences using proposional variables (P, Q, R, ...) and the logical connectives. State the translation keys: • Kate is a student. If Kate is a student, then Kate is broke. |= Kate is broke. • Kate is a student. Kate is broke. |= Kate is a student and Kate is broke. • Kate is a student and Kate is broke. |= Kate is a student. • Kate is a student. |= Kate is a student. • 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? • If Kate is a student, then Kate is broke. Kate is broke. (|= ??) Kate is a student.

  3. CA 208 Logic Ex1 Complete the following truth tables:

  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. 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. 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)

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