18 ~ Theory of Computer Science
EXAMPLE 1.19
Check the validity of the following argument:
If Ram has completed B.E. (Computer Science) or MBA, then he is
assured of a good job. If Ram is assured of a good job, he is happy. Ram is
not happy. So Ram has not completed MBA.
Solution
We can name the propositions in the following way:
P denotes 'Ram has completed B.E. (Computer Science)'.
Q denotes 'Ram has completed MBA'.
R denotes 'Ram is assured of a good job'.
S denotes 'Ram is happy'.
The given premises are:
(i) (P v Q) ~ R
(ii) R ~ S
(iii) --, S
The conclusion is --, Q.
1. (P v Q) ~ R
Premise (i)
2. R ~ S
Premise (ii)
3. (P v Q) ~ S
Lines 1, 2 and hypothetical syllogism RJ 7
4. --, S
Premise (iii)
5. --, (P v Q)
Lines 3, 4 and modus tollens RJ s
6. --, P /\ --, Q
DeMorgan's law h
7. --, Q
Line 6 and simplification RJ 3
Thus the argument is valid.
EXAMPLE 1.20
Test the validity of the following argument:
If milk is black then every cow is white. If every cow is white then it has
four legs. If every cow has four legs then every buffalo is white and brisk.
The milk is black.
Therefore, the buffalo is white.
Solution
We name the propositions in the following way:
P denotes 'The milk is black'.
Q denotes 'Every cow is white'.
R denotes 'Every cow has four legs'.
S denotes 'Every buffalo is white'.
T denotes 'Every buffalo is brisk'.
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