a)
Assume,
O(d) : I take the day(d) off.
R(d) : it rains on the day(d).
S(d) : it snows on the day(d).
s(d) : it was sunny on the day (d).
Premises:
1.$\forall_{d}$(O(d)$\rightarrow$(R(d)$\vee$S(d)))
2. O(Tuesday)$\vee$O(Thursday)
3. s(Tuesday) $\equiv$ $\neg$(R(Tuesday)$\vee$S(Tuesday))
4. $\neg$S(Thursday) $\equiv$ R(Thursday)
Using 1 by applying Universal Instantiation. If Tuesday is a particular element of the domain.we get,
O(Tuesday)$\rightarrow$(R(Tuesday)$\vee$S(Tuesday)) --------(5)
If Thursday is a particular element of the domain. we get,
O(Thursday)$\rightarrow$(R(Thursday)$\vee$S(Thursday)) --------(6)
Using 5 & 3 by applying modus tollens, we get $\neg$O(Tuesday) -------(7) .
Using 7 & 2 by applying Disjunctive syllogism, we get O(Thursday) ----------(8)
Hence, from 4, 7 and 8 we can conclude the conclusion is R(Thursday) ^ $\neg$O(Tuesday) ^ O(Thursday).
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b)
Assume,
E: I eat spicy foods.
D: I have strange dreams.
T: There is a thunder while I sleep.
Premises:
1. E$\rightarrow$D
2. T$\rightarrow$D
3. $\neg$D
Using 1 and 3 by applying modus tollens, we get $\neg$E -------(4)
Using 2 and 3 by applying modus tollens, we get $\neg$T --------(5)
Hence, from 4 and 5 we can conclude conclusion is $\neg$E ^ $\neg$T.
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c)
Assume,
C: I am clever.
L: I am lucky.
W: I will win the lottery.
Premises:
1. C$\vee$L
2. $\neg$L
3. L$\rightarrow$W $\equiv$ $\neg$L$\vee$W
Using 1 & 2 by applying disjunctive syllogism, we get C -------(4)
Using 1 & 3 by applying Resolution rule, we get C$\vee$W -----(5)
Hence, conclusion is C$\vee$W.
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d)
Assume,
C(x): x is a computer science major.
P(x): x has a personal computer.
Premises:
1. $\forall_{x}$(C(x)$\rightarrow$P(x))
2. $\neg$P(Ralph)
3. P(Ann)
Using 1 by applying universal Instantiation, if Ralph is a particular element of the domain.we get C(Ralph)$\rightarrow$P(Ralph) --------(4)
Using 4 and 2 by applying $\neg$C(Ralph)
Hence, we can conclude the conclusion is $\neg$C(Ralph) .
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e)
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f)
Assume,
R(x): x is a Rodent.
G(x): x gnaw their food.
Premises:
1. $\forall_{x}$(R(x)$\rightarrow$G(x))
2. R(Mice)
3. $\neg$G(Rabbit)
4. $\neg$R(Bats)
Using 1 by applying Universal Instantiation. If Mice is a particular element of the domain, we get R(Mice)$\rightarrow$G(Mice) -----(5)
If Rabbit is a particular element of the domain,we get R(Rabbit)$\rightarrow$G(Rabbit) -----(6)
Using 2 & 5 by applying modus ponnes, we get G(Mice) -----(7)
Using 3 & 6 by applying modus tollens, we get $\neg$R(Rabbit) -----(8)
Hence, from 7 and 8 we can conclude the conclusion is G(Mice) ^ $\neg$R(Rabbit).