Math Problem Statement
P or Q , Q then R , P then M , negation M prove that primises conclusion is P and (P or Q) in discrete mathematics with using simple rules
Solution
To prove that the premises lead to the conclusion , we will go through the given premises step by step using basic logical rules such as Modus Ponens, Disjunction Elimination, and others. The premises are:
- (Premise 1)
- (Premise 2)
- (Premise 3)
- (Premise 4)
We need to derive the conclusion .
Step-by-step derivation:
-
Restate the premises:
- From Premise 1, , we know that either or is true.
- Premise 2 tells us that if is true, then is true.
- Premise 3 states that if is true, then is true.
- Premise 4 tells us that is false (i.e., ).
-
Using and :
- Premise 4 gives us .
- From Premise 3, , using Modus Tollens (if and , then ), we can conclude:
-
From , deduce :
- From (Premise 1), since , we must conclude that is true (by Disjunction Elimination).
-
From , apply :
- Now that we know is true, we can use Premise 2, , and apply Modus Ponens to deduce:
-
Show the conclusion :
- Now, let’s check what we need for the conclusion .
- The first part of the conclusion is , which we don’t have (we actually have ). So, we cannot directly prove is true.
- However, for the second part, , we already know from Premise 1 that is true.
Therefore, this argument does not seem to support the conclusion as is, because we derived .
Conclusion:
The premises given do not directly lead to the conclusion . Instead, the steps show that is false and is true, which conflicts with the desired conclusion. Therefore, the conclusion cannot be validly inferred from the premises as written.
Do you want more details on any part of this explanation, or further clarification?
Relative questions:
- How does Modus Tollens apply in logical proofs?
- Can the conclusion be valid with other premises?
- How do Disjunction Elimination and Modus Ponens differ in logical reasoning?
- What are the common fallacies in reasoning with implications?
- How does the negation of a premise affect the validity of a conclusion?
Tip: Always identify whether the premises lead to a direct contradiction before attempting a proof.
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Math Problem Analysis
Mathematical Concepts
Propositional Logic
Logical Implication
Modus Ponens
Modus Tollens
Disjunction Elimination
Formulas
Modus Ponens: If P → Q and P is true, then Q is true
Modus Tollens: If P → Q and Q is false, then P is false
Disjunction Elimination: If P ∨ Q and ¬P, then Q
Theorems
Modus Ponens
Modus Tollens
Disjunction Elimination
Suitable Grade Level
Undergraduate (Discrete Mathematics)
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