Simple Logic Truth Table Needs Checking

In summary: I think I will just wing it and hope for the best. In summary, the professor thinks the implication is not an associative connective.
  • #1
Hotsuma
41
0
Dear All,

Having trouble with a seemingly simple logic truth table. Are these answers correct?

[tex]\begin{tabular}{| c | c | c | c | c | c |}
\hline
p & q & r & (p \vee q)\wedge(q \vee r) & (\neg p \wedge q) \vee ( p \wedge \neg r) & p \rightarrow q \rightarrow r\\
\hline
T & T & T & T & F & T \\
T & T & F & T & T & F \\
T & F & T & T & F & T \\
T & F & F & F & T & F \\
F & T & T & T & T & T \\
F & T & F & T & T & F \\
F & F & T & F & F & T \\
F & F & F & F & F & T \\
\hline
\end{tabular}[/tex]
 
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  • #2
Hi Hotsuma, I think the and/or/not parts are ok, but for the last part I think it would be

[tex]
\begin{tabular}{| c | c | c | c }
\hline
p & q & r & p \rightarrow q \rightarrow r\\
\hline
T & T & T & T \\
T & T & F & F \\
T & F & T & F \\
T & F & F & F \\
F & T & T & T \\
F & T & F & T \\
F & F & T & T \\
F & F & F & T \\
\hline\end{tabular}[/tex]

This effectively has 2 tests, it first requires that if p then q, if this is satisfied the the further condition that if q then r must be satisfied

In the 3rd case, as p is T and q is F, then the line is FALSE as the first condition is not satisfied

For the last 4 cases as p is F then the line is T by default, and we don't go any further - its a vacuous truth

what do you think?
 
Last edited:
  • #3
Dear lanedance,

I considered this but wasn't sure if that was correct or not. What you described to me in your past post could be written as "if p then q" and "if q then r," which I don't think is the same thing, or rather, I'm not sure whether it is or is not. Thanks for the suggestion though, having someone else recommend what I was thinking is always appreciated.
 
  • #4
And I'd love to prove that the two are tautologies, but since I am not sure about the former, that doesn't really help me much :(.


[tex](p \rightarrow q) \wedge (q \rightarrow r) \equiv p \rightarrow q \rightarrow r [/tex]
 
  • #5
Hotsuma said:
Dear lanedance,

I considered this but wasn't sure if that was correct or not. What you described to me in your past post could be written as "if p then q" and "if q then r," which I don't think is the same thing, or rather, I'm not sure whether it is or is not.

I don't think is is exactly the same thing, take the case:
p = F
q = T
r = F

this gives the following:

[tex] p \rightarrow q [/tex]
True - vacuously as P is False

[tex] q \rightarrow r [/tex]
False

[tex] (p \rightarrow q) \wedge (q \rightarrow r) [/tex]
False

[tex] p \rightarrow q \rightarrow r [/tex]
True - vacuously as P is False
 
Last edited:
  • #6
There is a problem here. Implication is not an associative connective. i.e.

[tex](p \rightarrow q) \rightarrow r \not\equiv p \rightarrow (q \rightarrow r)[/tex]

so the meaning of [itex]p \rightarrow q \rightarrow r[/itex] is ambiguous.

(Consider the case where p = F, q = T, r = F.)

Unless there is an assumption about the order of execution of multiple implications, the last column could be evaluated differently (either TFTTTFTF or TFTTTTTT).

--Elucidus
 
  • #7
Yeah, that was my fear. I e-mailed the professor at like 4PM but he has not yet responded (nor do I expect him to).
 

1. What is a simple logic truth table?

A simple logic truth table is a table used to represent the relationship between two or more logical statements. It displays all possible combinations of inputs and their corresponding outputs, allowing for the evaluation of logical expressions.

2. How do you create a simple logic truth table?

To create a simple logic truth table, you must first identify the logical statements and their corresponding inputs. Then, list all possible combinations of inputs and determine the output for each combination based on the logical operators used in the statements. Finally, organize the inputs and outputs into a table format.

3. Why is it important to check a simple logic truth table?

It is important to check a simple logic truth table to ensure that the logical statements and operators are correct and accurately represent the relationship between the inputs and outputs. This helps to avoid errors and ensures that the logic being used is sound.

4. What are some common errors when creating a simple logic truth table?

Some common errors when creating a simple logic truth table include forgetting to list all possible combinations of inputs, using incorrect logical operators, and not properly organizing the inputs and outputs in the table.

5. How can you verify the accuracy of a simple logic truth table?

To verify the accuracy of a simple logic truth table, you can manually check each input and output combination to ensure that it follows the logical statements and operators used. You can also use a truth table generator tool to double-check your results.

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