SOLUTION: i have been having a hard time answering this rules of inference and rules of replacement, can you help me answer this?
1) ﬢ[P→ﬢ(Q۸R)]
ﬢ[P→ﬢ
___________ ____
Algebra.Com
Question 1164909: i have been having a hard time answering this rules of inference and rules of replacement, can you help me answer this?
1) ﬢ[P→ﬢ(Q۸R)]
ﬢ[P→ﬢ
________________
∴P۸(Q۸R)
2) 1. (P۷Q)→(R۸S)
2. P
_________________
∴ R
3) 1. (P۷Q) ۷ ﬢR
2. ﬢP ۸ ﬢQ
__________________
∴ﬢR ۷ S
Answer by CPhill(2189) (Show Source): You can put this solution on YOUR website!
Rules of inference and replacement can feel like a complex puzzle at first, but once you identify the "shape" of the statements, the rules become much clearer. Here are the step-by-step proofs for your three problems.
---
### 1) Prove:
**Given:**
| Step | Statement | Rule Used |
| --- | --- | --- |
| 1 | | Premise |
| 2 | | **Material Implication (Impl)** on 1 |
| 3 | | **De Morgan's (DM)** on 2 |
| 4 | | **Double Negation (DN)** on 3 |
**Explanation:** To break into a negated conditional, we first turn the "if-then" into an "or" statement, then use De Morgan's to distribute the negation.
---
### 2) Prove:
**Given:** 1.
2.
| Step | Statement | Rule Used |
| --- | --- | --- |
| 1 | | Premise |
| 2 | | Premise |
| 3 | | **Addition (Add)** on 2 |
| 4 | | **Modus Ponens (MP)** on 1, 3 |
| 5 | | **Simplification (Simp)** on 4 |
**Explanation:** Since we have , we can "add" anything to it using an "or" statement. This satisfies the left side of the conditional in Step 1, allowing us to extract the right side.
---
### 3) Prove:
**Given:**
1.
2.
| Step | Statement | Rule Used |
| --- | --- | --- |
| 1 | | Premise |
| 2 | | Premise |
| 3 | | **De Morgan's (DM)** on 2 |
| 4 | | **Disjunctive Syllogism (DS)** on 1, 3 |
| 5 | | **Addition (Add)** on 4 |
**Explanation:** In Step 3, we used De Morgan's to show that neither nor is true. This contradicts the first part of Premise 1, which forces to be true. Once we have , we can use Addition to add any variable (in this case, ).
---
### Logic Tool Reference
A helpful tip for these is to always look at your **Conclusion** first. If it has a "" (like in problem 3), you often only need to find one side of it and use **Addition**.
Would you like to try another set, or should we go deeper into how **De Morgan's Law** works?
RELATED QUESTIONS
I am having a hard time learning the rules of replacement. Can you help me solve this... (answered by solver91311)
can you help me solve this proof using the first 13 rules of inference please?... (answered by jim_thompson5910)
Please help me create this proof using the 18 rules of inference. I have been working on... (answered by jim_thompson5910)
Help! I need this badly. Use rules of inference and replacement. Thaaaanks!
P > A
X > (answered by math_helper)
Please help me answer this one natural proof CORRECTLY using the 8 rules of implication... (answered by robertb)
I can only use the rules of inference/replacement
1) R v ~T
2) ~T -> O
3) ~O... (answered by jim_thompson5910)
Can you please assist me with these problems? We are using the Rules of Inference and the (answered by jim_thompson5910)
1. I v (N&F)
2. I > F /F
using rules of inference & replacement
(answered by Solver92311)
Can someone please help me with this proof using the 18 rules of inference?
> is a... (answered by jim_thompson5910)