CA 208 Continuous Assessment 2006/7

1. CA 208 Continuous Assessment 2006/7 • Name: _______________________________ • Student Number: _______________________ • Signature: • Date:

2. CA 208 Continuous Assessment 2006/7 • Formalise the following arguments/inferences in Propositional Logic using proposional variables (P, Q, R, ...) and the logical connectives. Give the translation key. • 4 is even. If 4 is even, then 4 is devisible by 2. |= 4 is devisible by 2. • 3 is odd. 2 is even. |= 3 is odd and 2 is even. • 3 is odd and 2 is even. |= 2 is even. • 3 is odd. |= 2 is even or 3 is odd.

3. CA 208 Continuous Assessment 2006/7 Complete the following truth table:

4. CA 208 Continuous Assessment 2006/7 • Use the truth table method to show whether the following are tautologies, contingencies or contradictions: • (P  Q) • (P  Q) (P  Q) • Use the Boolean equivalences to show (i.e.rewrite) that the following are logically equivalent: • (Q  P)  (P  Q) • (P  (Q  P))  (P  Q)

5. CA 208 Continuous Assessment 2006/7 • Complete ........... the following definition of the syntax of Propositional Logic with negation, conjunction, disjunction, material implication and the bi-conditional: • Let Π be a (countably infinite ...) set of propositional variables Π = {A, B, C, ...} • If Φ Π, then Φ is a formula. • If Φ is a formula, then ............ is a formula • If Φ and Ψ are formulas, then ................ is a formula • If Φ and Ψ are formulas, then ................ is a formula • If Φ and Ψ are formulas, then ................ is a formula • If Φ and Ψ are formulas, then ................ is a formula • Nothing else is a formula.

6. CA 208 Continuous Assessment 2006/7 • Prove the following in the Natural Deduction Calculus for Propositional Logic: • {(C  B), (B  A)} |- (C  A) • {A, (A (BC)), (C(DE)), (B(FE))} |- E • {A, ((BA)  C)} |- C

7. CA 208 Continuous Assessment 2006/7 • Given the definition of the syntax of a language of First Order Predicate Logic (FOPL), with Pred = {like²,student¹}, CONST = {j,k}, VAR = {x,y,z}, which of the following are well-formed formulas (WWFs) and which are not? • like(j,k) • like(k) • like  student • (like  student) • student(k) • student(j,j,k) • (like(j,k)  like(k,j)) • x like(k,x) • y x like(x,y) • like(j,k)  j • k  j • like(j  k) • x (student(x)  like(k,x)) • x (student(x)  like(k,x))

8. CA 208 Continuous Assessment 2006/7 • Interpretation in a model: given our definition of the syntax and semantics of First Order Predicate Logic (FOPL), and a (specific) language of FOPL with CONST = {j,k,m}, VARS = {x,y,z} and PRED = {student¹, broke¹, like²} and the following model M = < U, > with U = {□, ◊, ○} and (j) = □,(m) = ◊, (k) = ○ ,(student) = {□, ○}, (broke) = {□, ○} (like) = {<□,□>,<○,□>,<◊,○>} and a variable assignment function g with g(x) = ○, g(y)= ○, g(z)= ○, compute the truth value of the following formulas relative to model M and variable assignment function g (i.e. compute which of the following are satisfied in M and g)? • like(j,j) • (like(j,j)  like(k,j)) • x (student(x)  broke(x))

9. CA 208 Continuous Assessment 2006/7 • Axiomatise (i.e. describe) the following situation in FOPL with CONST = {j,k,m}, VARS = {x,y,z} and PRED = {older²} • Kate is older than John. John older than Mary. • Nobody is older than themselves. • If x is older than y, and y is older than z, then x is older than z. • Translate the following into FOPL and prove the resulting formulas from the axiomatisation above in the Natural Deduction proof system for FOPL: • Kate is older than Mary. • John is not older than John.