This question regards the notion of "Tautology". i. Using truth tables, or in any other way, prove th compound propositions is a tautology. These implications are four of the most importan propositional logic. Each rule gives a conclusior from a set of hypotheses. As such, these rules a a correct proof. a. [(P ⇒ Q) ^ P] ⇒ Q. b. [(P⇒ Q) ^ (~ Q)] ⇒ (~ P). E c. [(P⇒ Q) ^ (Q ⇒ R)] ⇒ (P ⇒ R). d. [(P VQ) ^ (~ P)] ⇒ Q.
This question regards the notion of "Tautology". i. Using truth tables, or in any other way, prove th compound propositions is a tautology. These implications are four of the most importan propositional logic. Each rule gives a conclusior from a set of hypotheses. As such, these rules a a correct proof. a. [(P ⇒ Q) ^ P] ⇒ Q. b. [(P⇒ Q) ^ (~ Q)] ⇒ (~ P). E c. [(P⇒ Q) ^ (Q ⇒ R)] ⇒ (P ⇒ R). d. [(P VQ) ^ (~ P)] ⇒ Q.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![This question regards the notion of 'Tautology'.
i. Using truth tables, or in any other way, prove that each of the following
compound propositions is a tautology.
These implications are four of the most important "rules of inference " in
propositional logic. Each rule gives a conclusion which follows logically
from a set of hypotheses. As such, these rules are the building blocks of
a correct proof.
a. [(P ⇒ Q) ^ P] ⇒ Q.
b. [(P ⇒ Q) ^ (~ Q)] ⇒ (~ P).
c. [(P ⇒ Q) ^ (Q ⇒ R)] ⇒ (P ⇒ R).
d. [(PVQ) ^ (~ P)] ⇒ Q.
ii. For each one of the rules of inference in part (i) of this question give an
example of a "real life" situation in which the rule can be applied.
Example: Consider the following two statements:
"If there was no pop-quiz last Tuesday then there will be a pop-quiz
next Tuesday".
"There was no pop-quiz last Tuesday".
Assuming both of these statements to be true, I can apply the tautology
in (a) to deduce that "there will be a pop-quiz next Tuesday".
(Who are P and Q in my example? How was the tautology applied?
Make sure to shortly answer these questions when you provide your own
examples).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fce8827c4-1422-4db4-bd87-5e9d779d6c3b%2Fe311f149-9e06-4502-9970-0478ffeafbed%2F77d9q09_processed.png&w=3840&q=75)
Transcribed Image Text:This question regards the notion of 'Tautology'.
i. Using truth tables, or in any other way, prove that each of the following
compound propositions is a tautology.
These implications are four of the most important "rules of inference " in
propositional logic. Each rule gives a conclusion which follows logically
from a set of hypotheses. As such, these rules are the building blocks of
a correct proof.
a. [(P ⇒ Q) ^ P] ⇒ Q.
b. [(P ⇒ Q) ^ (~ Q)] ⇒ (~ P).
c. [(P ⇒ Q) ^ (Q ⇒ R)] ⇒ (P ⇒ R).
d. [(PVQ) ^ (~ P)] ⇒ Q.
ii. For each one of the rules of inference in part (i) of this question give an
example of a "real life" situation in which the rule can be applied.
Example: Consider the following two statements:
"If there was no pop-quiz last Tuesday then there will be a pop-quiz
next Tuesday".
"There was no pop-quiz last Tuesday".
Assuming both of these statements to be true, I can apply the tautology
in (a) to deduce that "there will be a pop-quiz next Tuesday".
(Who are P and Q in my example? How was the tautology applied?
Make sure to shortly answer these questions when you provide your own
examples).
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VIEWStep 2: Find truth table for a) and prove that this is tautology.
VIEWStep 3: Find truth table for b) and prove that this is tautology.
VIEWStep 4: Find truth table for c) and prove that this is tautology.
VIEWStep 5: Find truth table for d) and prove that this is tautology.
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![This question regards the notion of 'Tautology'.
i. Using truth tables, or in any other way, prove that each of the following
compound propositions is a tautology.
These implications are four of the most important "rules of inference " in
propositional logic. Each rule gives a conclusion which follows logically
from a set of hypotheses. As such, these rules are the building blocks of
a correct proof.
a. [(P ⇒ Q) ^ P] ⇒ Q.
b. [(P ⇒ Q) ^ (~ Q)] ⇒ (~ P).
c. [(P ⇒ Q) ^ (Q ⇒ R)] ⇒ (P ⇒ R).
d. [(PVQ) ^ (~ P)] ⇒ Q.
ii. For each one of the rules of inference in part (i) of this question give an
example of a "real life" situation in which the rule can be applied.
Example: Consider the following two statements:
"If there was no pop-quiz last Tuesday then there will be a pop-quiz
next Tuesday".
"There was no pop-quiz last Tuesday".
Assuming both of these statements to be true, I can apply the tautology
in (a) to deduce that "there will be a pop-quiz next Tuesday".
(Who are P and Q in my example? How was the tautology applied?
Make sure to shortly answer these questions when you provide your own
examples).](https://content.bartleby.com/qna-images/question/ce8827c4-1422-4db4-bd87-5e9d779d6c3b/0e7b8269-5d36-4aea-b7fc-98c19935e7a6/0gtnqh_processed.png)
Transcribed Image Text:This question regards the notion of 'Tautology'.
i. Using truth tables, or in any other way, prove that each of the following
compound propositions is a tautology.
These implications are four of the most important "rules of inference " in
propositional logic. Each rule gives a conclusion which follows logically
from a set of hypotheses. As such, these rules are the building blocks of
a correct proof.
a. [(P ⇒ Q) ^ P] ⇒ Q.
b. [(P ⇒ Q) ^ (~ Q)] ⇒ (~ P).
c. [(P ⇒ Q) ^ (Q ⇒ R)] ⇒ (P ⇒ R).
d. [(PVQ) ^ (~ P)] ⇒ Q.
ii. For each one of the rules of inference in part (i) of this question give an
example of a "real life" situation in which the rule can be applied.
Example: Consider the following two statements:
"If there was no pop-quiz last Tuesday then there will be a pop-quiz
next Tuesday".
"There was no pop-quiz last Tuesday".
Assuming both of these statements to be true, I can apply the tautology
in (a) to deduce that "there will be a pop-quiz next Tuesday".
(Who are P and Q in my example? How was the tautology applied?
Make sure to shortly answer these questions when you provide your own
examples).
Solution
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