Prove that the following propositions are tautologies using rules of inference. a) ̴ ((p˄q) ˄ ̴ (p˅q)) b) ̴ (p-->q) --> p
Prove that the following propositions are tautologies using rules of inference. a) ̴ ((p˄q) ˄ ̴ (p˅q)) b) ̴ (p-->q) --> p
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
Prove that the following propositions are tautologies using rules of inference.
a) ̴ ((p˄q) ˄ ̴ (p˅q))
b) ̴ (p-->q) --> p
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The answer provided did not use the rules of inference to prove that the propositions are tautologies which is what the original question asked. Can the answer to this problem be provided by using the rules of inference?
Transcribed Image Text:itions: Logic and Proofs
TABLE 1 Rules of Inference.
Rule of Inference
P
P-9
:. 9
-q
P-9
קר .
P-9
9-r
.. pr
PV q
P
:.qu
P
:: pv q
p^ q
.. P
P
9
:. p^g
pv q
pvr
:: qvr
Tautology
(p^(p-q)) → 9
(-q ^ (p →q))
((p →q) ^ (q→r)) → (p → r)
→ P
-
((pvq) ^-p) → 9
P→ (pv q)
(p^q) → р
((p) ^ (q)) → (p^q)
((pvq) ^ (p Vr)) → (qvr)
This is an argument that uses the addition rule.
Name
Modus ponens
Modus tollens
Hypothetical syllogism
Disjunctive syllogism
Addition
Simplification
Conjunction
Resolution
LE 4 State which rule of inference is the basis of the following argument: "It is below
raining now. Therefore, it is below freezing now."
Solution: Let p be the proposition "It is below freezing now," and let q be the prop
raining now." This argument is of the form si botellons de l
on biley sus
to estu
Solution
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