BMAN20072 Investment Analysis
Workshop 1 solution
Accounting and Finance Division
Manchester Business School
1. Assume the following
Investment
Expected return
Standard deviation
1
0.12
0.3
2
0.15
0.5
3
0.21
0.16
4
0.24
0.21
Based on utility value, which
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Predicate languages (all sections): Solutions
1. Let L be a predicate language with constant symbols c and d, unary function
symbol f , binary function symbol g , unary relation symbol P and ternary(=3ary) relation symbol R.
(a) Which of the following a
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A Hilbert-style system: Solutions
1. Show, using the Hilbert-style calculus (which should be used for all the
questions in this section), that p, q r (p q ) r. Now re-do this using
the Deduction Theorem as a derived rule of inference.
Solution
1. cfw_p,
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A natural deduction system: Solutions
1. Re-do Exercise 1 from the previous set using the natural deduction/Gentzenstyle calculus.
Solution
p, q r, p q p p, q r, p q p q
p, q r, p q q r
p, q r, p q q
p, q r, p q r
p, q r (p q ) r
The sequence of justica
6
Interpolation: Solutions
1. You are given that (p r) (q r) |= (s p) (s q ). Find an
interpolant between (p r) (q r) and (s p) (s q ) which involves
the common propositional variables, p, q , only.
Solution First well nd a disjunctive normal form for (p
1
Propositional terms: Solutions
1. Suppose that:
p is the proposition It is raining.
q is the proposition There are no clouds in the sky.
r is the proposition We are in Manchester.
s is the proposition We have no umbrella.
(a) Render into reasonable Engl
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Adequate sets of connectives: Solutions
1. Show that cfw_, is an adequate set of connectives.
Solution p q is logically equivalent to p q ; p q is equivalent to (p q );
p q is equivalent to (p q ) (q p)
2. The NOR connective has the truth table shown.
3
Beth trees: Solutions
1. Use Beth trees to determine whether or not each of the following is true:
(i) p q |= (p q ) q ;
(ii) p r, (q r) |= p q .
Solution (i)
p q | ( p q ) q
VVVV
ll
VVVV
lll
VVVV
lll
VVVV
lll
VVVV
ll
l
p q|q
p q | p q
HH
HH
tt
t
HH
tt
2
Valuations: Solutions
1. Draw up truth tables for each of the following propositional terms.
(i) (p q ) r;
(ii) p p;
(iii) p p;
(iv) (p q ) (q r) p);
(v) (p q ) r;
(vi) p q p;
(vii) (p q r) (p (q r); (viii) (p p) (q q )
Which are tautologies? Which are
4
Normal forms: Solutions
1. Find a disjunctive normal form for each of the following propositional terms:
(i) p (p q );
(ii) (p r) (q s);
(iii) (p q ) (q p);
(iv) (p (p q ) (p q );
(v) (p q ) (p (q r).
For each of the above also nd a propositional term i