Prove Logical Equivalence of P->(Q or R)

In summary, the statement (P -> Q) or (P -> R) is equivalent to P -> (Q or R). However, the steps used to prove this statement are incorrect. The correct proof would involve transforming (P -> Q) or (P -> R) into P -> (Q and R) and P -> (Q or R) into (P -> Q) and (P -> R).
  • #1
The Subject
32
0
From the text it says (P -> Q) or (P -> R) is equivalent to P -> (Q or R)

I tried to see if this is true so I tried
[tex] (P \to Q) \lor (P \to R) \\
(P \lor \neg Q) \lor (P \lor \neg R) \\
P \lor \neg Q \lor \neg R \\
P \lor \neg(Q \land R) \\
P \to (Q \land R) [/tex]
and
[tex] P \to (Q \lor R) \\
P \lor \neg(Q \lor R ) \\
P \lor (\neg Q \land \neg R) \\
(P \lor \neg Q) \land (P \lor \neg R) \\
(P \to Q) \land (P \to R) [/tex]

From what I've done its seems like they're not equivalent ?
 
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  • #2
The Subject said:
I tried
[tex] (P \to Q) \lor (P \to R) \\
(P \lor \neg Q) \lor (P \lor \neg R) [/tex]
The second line does not follow from the first.

I think what you meant to write for the second line was
$$(\neg P\vee Q)\vee (\neg P\vee R)$$
which is not the same thing.
 
  • #3
andrewkirk said:
The second line does not follow from the first.

I think what you meant to write for the second line was
$$(\neg P\vee Q)\vee (\neg P\vee R)$$
which is not the same thing.
AHHHH thank you!
 

1. What is "Prove Logical Equivalence of P->(Q or R)"?

"Prove Logical Equivalence of P->(Q or R)" is a statement in propositional logic that asks to show that the logical implication, P implies (Q or R), is equivalent to the logical disjunction of the negation of P and the disjunction of Q and R.

2. How do you prove logical equivalence in propositional logic?

To prove logical equivalence in propositional logic, you need to use logical equivalences, such as the commutative, associative, and distributive properties, in combination with the rules of inference, such as modus ponens and modus tollens, to show that two statements have the same truth values for all possible truth assignments to their propositional variables.

3. What is the difference between logical equivalence and material equivalence?

Logical equivalence refers to the relationship between two statements in propositional logic where they have the same truth values for all possible truth assignments. Material equivalence, on the other hand, refers to the relationship between two statements in propositional logic where they have the same truth values for a particular truth assignment.

4. Can you give an example of proving logical equivalence in propositional logic?

Yes, for example, to prove the logical equivalence of P->(Q or R) and (~P v (Q v R)), we can use the distributive property to rewrite the statement as (~P v Q) v (~P v R) and then use the associative property to show that it is equivalent to (~P v ~P) v (Q v R), which simplifies to ~P v (Q v R). This shows that the two statements are logically equivalent.

5. Why is proving logical equivalence important in science?

Proving logical equivalence is important in science because it allows us to show that two statements or theories are equivalent and can be used interchangeably. This helps us to simplify complex ideas and make logical deductions, which are crucial in scientific research and experimentation.

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