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 P(PQ)P \wedge (P \vee Q), 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:

  1. PQP \vee Q (Premise 1)
  2. QRQ \rightarrow R (Premise 2)
  3. PMP \rightarrow M (Premise 3)
  4. ¬M\neg M (Premise 4)

We need to derive the conclusion P(PQ)P \wedge (P \vee Q).


Step-by-step derivation:

  1. Restate the premises:

    • From Premise 1, PQP \vee Q, we know that either PP or QQ is true.
    • Premise 2 tells us that if QQ is true, then RR is true.
    • Premise 3 states that if PP is true, then MM is true.
    • Premise 4 tells us that MM is false (i.e., ¬M\neg M).
  2. Using ¬M\neg M and PMP \rightarrow M:

    • Premise 4 gives us ¬M\neg M.
    • From Premise 3, PMP \rightarrow M, using Modus Tollens (if PMP \rightarrow M and ¬M\neg M, then ¬P\neg P), we can conclude: ¬P\neg P
  3. From ¬P\neg P, deduce QQ:

    • From PQP \vee Q (Premise 1), since ¬P\neg P, we must conclude that QQ is true (by Disjunction Elimination). QQ
  4. From QQ, apply QRQ \rightarrow R:

    • Now that we know QQ is true, we can use Premise 2, QRQ \rightarrow R, and apply Modus Ponens to deduce: RR
  5. Show the conclusion P(PQ)P \wedge (P \vee Q):

    • Now, let’s check what we need for the conclusion P(PQ)P \wedge (P \vee Q).
    • The first part of the conclusion is PP, which we don’t have (we actually have ¬P\neg P). So, we cannot directly prove PP is true.
    • However, for the second part, PQP \vee Q, we already know from Premise 1 that PQP \vee Q is true.

    Therefore, this argument does not seem to support the conclusion P(PQ)P \wedge (P \vee Q) as is, because we derived ¬P\neg P.


Conclusion:

The premises given do not directly lead to the conclusion P(PQ)P \wedge (P \vee Q). Instead, the steps show that PP is false and QQ 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:

  1. How does Modus Tollens apply in logical proofs?
  2. Can the conclusion P(PQ)P \wedge (P \vee Q) be valid with other premises?
  3. How do Disjunction Elimination and Modus Ponens differ in logical reasoning?
  4. What are the common fallacies in reasoning with implications?
  5. 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)