Discrete Mathematics - Reduction/Deduction!

[MogiYagi]

Hi!

I'm having some trouble understanding how I'm supposed to use the reduction and deduction methods.
The question is:

(p → q) ∧ (¬p → r) ∧ ((¬p ∧ r) → s) ∧ ¬q ⇒ s

Prove that this is correct, with the deduction AND reduction method.

I'm fairly new to this kind mathematics, so if somebody could explain and guide me through the answer I would be really grateful!
PS: Excuse the bad grammar (if any), english is my fourth language.

Thanks in advance.

Greetings!

 

[stapel]

You can view a similar discussion here.

Please note that, there also, the precise meaning of "deduction method" and "reduction method" is unknown by the helpers. It might be useful if you replied here with the definitions for these methods. Thank you!

 

[pka]

I'm supposed to use the reduction and deduction methods.
The question is:
(p → q) ∧ (¬p → r) ∧ ((¬p ∧ r) → s) ∧ ¬q ⇒ s
Prove that this is correct, with the deduction AND reduction method.

You are given that \(\displaystyle \left( {p \to q} \right) \wedge (\neg p \to r) \wedge \left[ {(\neg p \wedge r) \to s} \right] \wedge \neg q\).
As noted in the reply above, authors all have different names for logical operations. I follow Copi in general.
That statement has three conjunctions, by simplification we get \(\displaystyle \left( {p \to q} \right)\wedge \neg q\)

Modus tollens(MT) gives you \(\displaystyle \neg p\). Again simplification gives \(\displaystyle (\neg p\to r)\) and modus ponens(MP) gives \(\displaystyle r\).

Then addition \(\displaystyle (\neg p\wedge r)\) gives \(\displaystyle s\).

Now, if you do not know that vocabulary then look up the terms.

 

[MogiYagi]

Hi guys!

Thanks for the replies!

I've done the following:

(p → q) ∧ (¬p → r) ∧ ((¬p ∧ r) → s) ∧ ¬q ⇒ s

1. ¬q (condition)
2. p → q (-II-)
3. ¬p (Modus Tollens)
4. ¬p → r (condition)
5. pvr (equivalence) <=> ¬p ∧ r
6. r (disjunctive syllogism)
7. ¬p ∧ r (conjunction)
8. (¬p ∧ r) → s (condition)
9. s (Modus ponens)

1. q can not be true because ¬q is true.
2. s is false, which means ¬p ∧ r is also false.
3. p is false because q is false.
4. r is true because ¬p is true.
5. ((¬p ∧ r) → is false, which gives us an divergence.

(p → q) ∧ (¬p → r) ∧ ((¬p ∧ r) → s) ∧ ¬q ⇒ s
5. 2. 2. 6. 4. 2. 2.3.1.2
0. 1. 1. 1. 0. 1. 1.0.0.0

To all of this I got the following response:
- Adoption missing.
- The figures in the expression can not keep up with your numbers in the text below.
- Not clear why point 5 and what is the contradiction.
- Conclusion (what is contradicted really?)

Could somebody tell me how I can "fix" these problems. I'm kinda new to the reduction- and deduction methods so I was basically using the notes from my lectures and Google as a reference. Probably why I managed to screw things up. But it looks like I'm halfway to finishing it? Where did I go wrong in my thinking?

Thanks in advance!

Greetings!

 

[stapel]

It would really help if you could define what your book means by the "reduction" and "deduction" methods. Also, what does your instructor mean by "adoption"? What are the numbers for? What do they mean, and in what sense are they supposed to "keep up" with... something?