Comp232 - Mathematics For Computer Science - Ali Moallemi, Iraj Hedayati - Concordia University Page 16
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18Exercise 29
Use rules of inference to show that if x (P (x)
Q(x)),
x ( Q(x)
S(x)), x (R(x)
S(x)), and x P (x) are true, then
x R(x) is true.
Answer:
1)
x P (x) Premise
2)
P (c) Existential instantiation using (1)
3)
x (P (x)
Q(x)) Premise
4)
P (c)
Q(c) Universal instantiation using (3)
5)
Q(c) Disjunctive syllogism using (4) and (2)
6)
x ( Q(x)
S(x)) Premise
7)
Q(c)
S(c) Universal instantiation using (6)
8)
S(c) Disjunctive syllogism using (5) and (7)
9)
x (R(x)
S(x)) Premise
10)
R(c)
S(c) Universal instantiation using (9)
11)
R(c) Modus tollens using (8) and (10)
12)
x R(x) Existential generalization using (11)
Ali Moallemi, Iraj Hedayati
COMP232 - Mathematics for Computer Science
16 / 18
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