Mathematical Foundation of Computer Science

DHARM

118 MATHEMATICAL FOUNDATION OF COMPUTER SCIENCE

Thus, for following truthvalues, argument is invalid,

A:T

B:F

C : ......(T/F)

D:F

E:F

F : .......(T/F)

G:F

H:F

Example 5.12. For what (truth) values of V, H and O following argument is invalid.
(i) V → O
(ii) H → O
∴ V → H

Sol. Form the truth table shown in Fig 5.22 for the given argument; we find one such condition
s.t. V → H is T and H → O is also T and conclusion V → H is F, so the argument is invalid. We
also observe from the truth table shown in Fig. 5.22 that the conclusion is F and premises are
both T when V is T, H is T and O is T.

VHOV → OH → OV → H

FFFTTT

FFTTTT

FTFTFT

FTTTTT

TFFFTF

TFT T T F

TTFFFT

TTTTTT

Fig. 5.22

Example 5.13. Justify the validity of the argument.
“If prices fall then sell will increase; if sell will increase then Stephen makes whole money.
But Stephen does not make whole money; therefore prices are not fall.”
Sol. Represent the statement into the symbolic form,
(i)P → S (by assuming) prices falls : P
(ii)S → J sell will increase : S
(iii) ~ J John makes whole money : J
∴ ~ P
From the given premises & conclusion we obtain the formula i.e.,
(((P → S) ∧ (S → J) ∧ ~ J) → ~ P) : (say) X

R

S|

T|