Equivalence of (P -> R) V (Q -> R) and (P ∧ Q) -> R

In summary, the equivalence (P -> R) V (Q -> R) can be formulated as (¬P ∧ ¬Q) V R, which is equivalent to ¬(P ∧ Q) V R and (P ∧ Q) -> R. This formulation may seem counter-intuitive, but it can be understood by considering the truth tables for both expressions. Additionally, there was a typo in the third line of the original conversation, which was corrected to show the correct formulation.
  • #1
Dysnex
2
0
I can see how the equivalence can formulated with

(P -> R) V (Q -> R)
= (¬P V R) V (¬Q V R)
= (¬P ∧ ¬Q) V R
= ¬(P ∧ Q) V R
= (P ∧ Q) -> R
(Sorry, I would've written this in LaTeX if I were more competent.)

although I still it counter-intuitive and, at a glance, first thought it was (P V Q) -> R. I asked someone else and they also arrived at (P V Q) -> R, which seems to contradict(?) the above formulation and the book's answer key. Am I missing something?

Any help would be appreciated, thanks!
 
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  • #2
You have a typo on the third line: it's supposed to be "(¬P V ¬Q) V R" and then by DeMorgan's rule you get the 4th line ¬(P ∧ Q) V R. Maybe that was bothering you?

As for the intuitiveness of it. Think about when any of (P -> R) V (Q -> R) and (P ∧ Q) -> R are false: only when both P and Q are true but R is false; in both expressions. In all the other cases both expressions are true simultaneously. Maybe if you write truth tables for them it would be easier to see it.
 
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  • #3
If p or q are false the statement (P^Q)-> R will always be true. The same can be said for the statement (P -> R) V (Q -> R). If the first statement in P->Q is false then P ->Q will always be true.
 
  • #4
tauon is right about the typo. Adding a step or so will reveal why:

Original -

(¬P V R) V (¬Q V R)
= (¬P ∧ ¬Q) V R

New -

(¬P V R) V (¬Q V R)
(¬P V (R V ¬Q) V R) Association
(¬P V (~Q V R) V R) Commutation
(¬P V ~Q) V (R V R) Association
(¬P V ~Q) V (R) Idempotence

The conjunct doesn't even enter into this up to here. Now, go on to DeMorgan etc., as tauon says.
 
  • #5


I can confirm that your formulation of the equivalence is correct. The key to understanding this equivalence is to remember that the logical operator "or" (V) is inclusive, meaning that if either statement is true, the entire statement is true. This means that in the first statement (P -> R) V (Q -> R), if either P or Q is true, then R must also be true in order for the statement to be true. In the second statement, (P ∧ Q) -> R, both P and Q must be true in order for R to be true. This is why the two statements are equivalent - they both express that if either P or Q (or both) are true, then R must be true.

It is understandable that the formulation may seem counter-intuitive at first glance, especially if you are used to thinking of "or" as strictly meaning "either this or that". However, in logic, "or" has a broader meaning and can encompass the possibility of both statements being true. I would recommend practicing with more examples to become more comfortable with this concept. Remember, as scientists, we must always be open to new ways of thinking and approaching problems. Good luck!
 

1. What is the meaning of "equivalence" in this statement?

In this statement, "equivalence" refers to the logical concept that two statements have the same truth value, meaning they are either both true or both false.

2. Can you provide an example of the equivalence of (P -> R) V (Q -> R) and (P ∧ Q) -> R?

One example is the statement "If it rains, I will bring an umbrella" (P -> R) or "If it snows, I will bring an umbrella" (Q -> R). These can be combined to create the statement "If it rains and snows, I will bring an umbrella" (P ∧ Q) -> R. Both statements have the same truth value as they both result in bringing an umbrella if it both rains and snows.

3. How can I prove the equivalence of (P -> R) V (Q -> R) and (P ∧ Q) -> R?

This can be proven using a truth table, where all possible combinations of truth values for P and Q are tested. If the resulting truth values for (P -> R) V (Q -> R) and (P ∧ Q) -> R are the same for each combination, then the statements are equivalent.

4. Is this statement always true, or are there exceptions?

This statement is always true as it is a logical equivalence, meaning it holds true for all possible truth values of P and Q.

5. How can the equivalence of (P -> R) V (Q -> R) and (P ∧ Q) -> R be applied in real-life situations?

This equivalence can be applied in various fields such as mathematics, computer science, and philosophy. In mathematics, it can be used to simplify logical expressions and proofs. In computer science, it can be used in programming and circuit design. In philosophy, it can be used to analyze arguments and logical reasoning.

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