York University
‐
CSE 3401
‐
Winter 2012
Page: 1
York University
‐
Department of Computer Science and Engineering
SC/CSE 3401 3.00 – Functional and Logic Programming
Solutions to assignment 1
1) (8 marks) Consider the following formulae:
)
(
:
:
)
(
:
r
p
q
C
r
q
B
r
q
p
A
a)
Using truth tables, which of the above formulae is satisfiable?
b)
Which of the above formulae is a contradiction?
c)
Which of the following sets are satisfiable?
i.
{A, B}
ii.
{B, C}
iii.
{A, B, C}
d)
Which of the above sets i
‐
iii are inconsistent?
Explain your answers.
Answer.
Showing state values true as 1 and false as 0, the truth tables for formula A, B, and C is:
p
q
r
r
q
A:
)
(
r
q
p
B:
r
q
q
r
p
C:
)
(
r
p
q
1
0
0
0
0
1
1
1
0
0
2
0
0
1
1
1
0
1
1
1
3
0
1
0
1
1
0
0
0
0
4
0
1
1
1
1
1
0
1
0
5
1
0
0
0
0
1
1
1
1
6
1
0
1
1
1
0
1
1
1
7
1
1
0
1
1
0
0
1
0
8
1
1
1
1
1
1
0
1
0
a)
All 3 formulae, A, B, and C, are satisfiable since there is at least one row which makes them true.
For example row 1 shows a state which makes formula A true. Row 4 is a possible state that can
make formula B true. Row 5 is a possible state than can make formula C true.
b)
None of the 3 formula A, B, or C is a contradiction, since all of them are satisfiable.
c)
For the sets:
i.
{A,B} is satisfiable. There exists a state (for example row 1) that makes both formulae true.
ii.
{B, C} is satisfiable. There exists a state (row 5) that makes both formulae true.
iii.
{A, B, C} is not satisfiable. There is no state that can make all three formulae true.

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