Transcript CA 208 Logic - DCU School of Computing

CA 208 Continuous Assessment 2006/7
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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.
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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.
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CA 208 Continuous Assessment 2006/7
Complete the following truth table:
P
Q
1
1
1
0
0
1
0
0
PQ
PQ
PQ
PQ
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CA 208 Continuous Assessment 2006/7
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Use the truth table method to show whether the
following are tautologies, contingencies or
contradictions:
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(P  Q)
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(P  Q)  (P  Q)
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Use the Boolean equivalences to show (i.e.rewrite) that
the following are logically equivalent:
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(Q  P)  (P  Q)
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(P  (Q  P))  (P  Q)
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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:
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Let Π be a (countably infinite ...) set of propositional variables Π =
{A, B, C, ...}
If Φ  Π, then Φ is a formula.
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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
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Nothing else is a formula.
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CA 208 Continuous Assessment 2006/7
Prove the following in the Natural Deduction
Calculus for Propositional Logic:
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{(C  B), (B  A)} |- (C  A)
{A, (A (BC)), (C(DE)), (B(FE))} |- E
{A, ((BA)  C)} |- C
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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?
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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))
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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)?
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like(j,j)
(like(j,j)  like(k,j))
x (student(x)  broke(x))
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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}
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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:
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Kate is older than Mary.
John is not older than John.
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