Use inference rules to deduce the following conclusions from the following sets of premises: a) Premises: p ∨ q q → r, p ∧ s → t, ¬r, ¬q → u ∧ s Conclusion: t b) Premises: (p ∧ t) → (r ∨ s), q → (u ∧ t), u → p, ¬s Conclusion: q → r
The Answer to the Question
is below this banner.
Can't find a solution anywhere?
NEED A FAST ANSWER TO ANY QUESTION OR ASSIGNMENT?
Get the Answers Now!You will get a detailed answer to your question or assignment in the shortest time possible.
Here's the Solution to this Question
a. Solution for - Premises: p ∨ q, q → r, p ∧ s → t, ¬r, ¬q → u ∧ s Conclusion: t
1 | q → r | Premise |
2 | ¬ r | Premise |
3 | ¬ q | Modus Tollens of 1 and 2 |
4 | p ∨ q | Premise |
5 | p | Disjunctive syllogism of 5 and 4 |
6 | ¬q → u ∧ s | Premise |
7 | ¬ q | Copy 3 |
8 | u ∧ s | Modus Ponens of 7 and 6 |
9 | s | Simplification of 8 |
10 | p | Copy 5 |
11 | p ∧ s | Conjunction of 9 and 10 |
12 | p ∧ s → t | Premise |
13 | t | Modus Ponens of 11 and 12 |
By arriving at the conclusion "t", the argument is proven.
∴ The premise and conclusion is valid.
b. Solution for - Premises: (p ∧ t) → (r ∨ s), q → (u ∧ t), u → p, ¬s Conclusion: q → r
Since the above argument includes a proposition "q", the argument above will be valid, if the argument with the following premise and conclusion below is valid.
Knowing this, we can rewrite
Premises: (p ∧ t) → (r ∨ s)
q → (u ∧ t)
u → p
¬s
Conclusion: q → r
as
Premise: (p ∧ t) → (r ∨ s)
q → (u ∧ t)
u → p
¬s
q
Conclusion: r
Observe the two propositions and how they differ from each other. If we can prove that the second proposition is valid, it automatically means that the first (original) proposition is also valid.
1 | q → (u ∧ t) | Premise |
2 | q → u | Simplification of 1 |
3 | u → p | Premise |
4 | q → p | Hypothetical Syllogism of 2 and 3 |
5 | q → t | Simplication of 1 |
6 | q → (p ∧ t) | Conjunction of 4 and 5 |
7 | (p ∧ t) → (r ∨ s) | Premise |
8 | q → (r ∨ s) | Hypothetical Syllogism of 6 and 7 |
9 | q | Premise |
10 | r ∨ s | Modus Ponens of 8 and 9 |
11 | ¬s | Premise |
12 | r | Disjunctive syllogism of 10 and 11 |