-
For the second row, to determine the numbers of Trues (Ts) and Falses (Fs), we use the formula,
2n
2
2
=
23
2
2
=
8
2
4
= 2 = 2. Now, what this means is two alternating T and F. So,
2
R
T
T
T
T
F
F
F
F
-
G
T
T
F
F
T
T
F
F
2n
2
2
2
=
23
2
2
2
==
R
T
T
T
T
F
F
F
F
R ⊃ (G ∨ S)
8
2
2
2
2
=
4
2
2
2n
2
2
2
.
2
= 2 = 1. Now, what this means is one alternating T and F. So,
G
T
T
F
F
T
T
F
F
S
T
F
T
F
T
F
T
F
G∨S
R ⊃ (G ∨ S)
Now, let’s have G ∨ S. So, we don’t mind R here, we just look at G and S. And since G ∨ S is a
Disjunction, the rule is a Disjunction is True if at least one member of the Disjunction is True. It’s
False if both are False. So, in this case,
R
T
T
T
T
F
F
F
F
-
G∨S
For the third row, to determine the numbers of Trues (Ts) and Falses (Fs), we use the formula,
So,
-
S
G
T
T
F
F
T
T
F
F
S
T
F
T
F
T
F
T
F
G∨S
T
T
T
F
T
T
T
F
R ⊃ (G ∨ S)
So, after we’re done with G ∨ S, we’re ready to write the whole Truth Value for R ⊃ (G ∨ S). So,
what we need to look at here is the R column and the G ∨ S column, which is your Consequent. So,
again, your Conditional is False if your Antecedent is True and its Consequent is False, for all other
instances it’s True. So, the only row we have this is in the second row. Thus,