(a) If my program runs, then I am happy. If I am happy, the sun
shines. It’s 11.00 p.m. and very dark.
(b) All trignometric functions are periodic functions and all periodic
functions are continuous functions.
Solu tion
P
H
S
: My program runs
: I am happy
: Sun shines
: It' s
τ
11 pm.
: It is very dark
Hypothese
¬
→
→
∧ ¬



 
S
P
H
H
S
C
S
s
1.
P
H
→
P
2.
H
S
→
P
3.
P
S
→
(1), (2)
4.
τ ∧ ¬ S
P
5.
¬ ∧
S C
6.
¬ S
Sim pli fi ca tion
7.
¬ P
(3), (6), Modus Tollens
8.
¬ H
(1), (7), Modus Tollens
9.
¬ ∧ ¬
P
H
Con junc tion
Conclusion: I am not happy and my program does not run.
(b) T : Trignometric functions
P : Periodic functions
C : Continuous functions
T
P
P C
T
C
→
→
∴ →
Conclusion: All Trignometric functions are continuous functions.
Ì Exam ple 8.2.10: Construct a proof for each of the following
arguments, giving all necessary additional assertions. Specify the rules of
inference used at each step. (The word “or” denotes the “logical or” rather
than the “exclusive or”.
(a) It is not the case that IBM or Xerox will take over the copier
market. If RCA returns to the computer market, then IBM will
take over the copier market. Hence, RCA will not return to the
computer market.
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Theory of Automata, Formal Languages and Computation