Fill in the missing parts for the proof of validity of the following argument: • All students in this class who likes pineapple either likes mangoes or likes rambutan. • Every student in this class do not like mangoes. • There is a student in this class who doesn't like rambutan. • Therefore, there is a student in this class who doesn't like pineapple. Let the following symbols represent the corresponding predicates: P(x) -x likes pineapples M(x) -x likes mangoes R(x) -x likes rambutan where x is a student in the class. Thus, in symbolic form, we have V±(P(x) → (M(x) v R(1))) VI(~ M(I)) (~ R(x)) ..(~ P(x)) A proof of validity: step statement 1 2 3 4 5 6 7 8 9 10 VI(P(1) › (M(z) v R(x))) Vz(~ M(z)) (~R(1)) N ÷ M(a) for some element a ÷ P(a) → (M(a) v R(a)) P(a) (~ P(x)) reason premise premise premise 3, existential instantiation 2₁ 5,4, conjunction 6, De Morgan's 1, universal instantiation + 8,7, + 9, existential generalization
Fill in the missing parts for the proof of validity of the following argument: • All students in this class who likes pineapple either likes mangoes or likes rambutan. • Every student in this class do not like mangoes. • There is a student in this class who doesn't like rambutan. • Therefore, there is a student in this class who doesn't like pineapple. Let the following symbols represent the corresponding predicates: P(x) -x likes pineapples M(x) -x likes mangoes R(x) -x likes rambutan where x is a student in the class. Thus, in symbolic form, we have V±(P(x) → (M(x) v R(1))) VI(~ M(I)) (~ R(x)) ..(~ P(x)) A proof of validity: step statement 1 2 3 4 5 6 7 8 9 10 VI(P(1) › (M(z) v R(x))) Vz(~ M(z)) (~R(1)) N ÷ M(a) for some element a ÷ P(a) → (M(a) v R(a)) P(a) (~ P(x)) reason premise premise premise 3, existential instantiation 2₁ 5,4, conjunction 6, De Morgan's 1, universal instantiation + 8,7, + 9, existential generalization
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Transcribed Image Text:Fill in the missing parts for the proof of validity of the following argument:
• All students in this class who likes pineapple either likes mangoes or likes rambutan.
• Every student in this class do not like mangoes.
• There is a student in this class who doesn't like rambutan.
• Therefore, there is a student in this class who doesn't like pineapple.
Let the following symbols represent the corresponding predicates:
P(x) -x likes pineapples
M(x) -x likes mangoes
R(x) -x likes rambutan
where x is a student in the class.
Thus, in symbolic form, we have
V±(P(x) → (M(x) v R(1)))
VI(~ M(I))
(~ R(x))
..(~ P(x))
A proof of validity:
step statement
1
2
3
4
5
6
7
8
9
10
VI(P(1) › (M(z) v R(x)))
Vz(~ M(z))
(~R(1))
N
÷
M(a)
for some element a
÷
P(a) → (M(a) v R(a))
P(a)
(~ P(x))
reason
premise
premise
premise
3, existential instantiation
2₁
5,4, conjunction
6, De Morgan's
1, universal instantiation
+
+
8,7,
9, existential generalization
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