Truth table / logic

[love/hatewithMath]

Good to be back. This time, I'm comparing a truth table that I drafted to my textbook solution. First, I'll show them both. Then, I'll tell you what it is about them that I don't understand:



What I don't get about this is that the textbook solution, the table on top in the image above, omits scenarios in which p is FALSE and t is FALSE and the one in which p is TRUE and t is FALSE. Why are these scenarios unnecessary? Are they impossible, unuseful, or something else?

 

[JeffM]

I don't like what is in your text book.

So long as p and t are arbitrary propositions and it is understood that T v F and F v T are equivalent, there is no loss of generality in just showing three

T v T is T
T v F is T
F v F is F

but it does raise a question in the student's mind about that equivalence. The whole point of truth tables is to answer questions in the student's mind. Now all this may be irrelevant in the context of a specific problem, but you have not given us a specific problem

 

[pka]

Please have a careful look at THIS LINK That is the way truth tables should look.
You can use that to check your work. Here is another.
Note how the table is constructed. Three columns and \(2^3=8\) rows

 

[Steven G]

You gave us the textbook's solution, now can you give us the question from the textbook? Maybe there is an obvious reason why some rows were omitted.

 

[love/hatewithMath]

can you give us the question from the textbook?

Sure. Here it is:

"Determine whether the statement forms are logically equivalent. In each case, construct a truth table and include a sentence justifying your answer.

p v t and t"

In addition to the table, the textbook solution adds, "p v t and t always have the same truth values, so they are logically equivalent (This proves one of the universal bound laws.)."

 

[pka]

I think that if you read carefully, your text mean the \(\large t\) is a statement which is always true. such as \(2+2=4\).
Thus \(\begin{array}{*{20}{c}} p&t&{p \vee t} \\ \hline T&T&T \\ F&T&T \end{array}\)
So that proves that if \(p\) is any statement and \(t\) is a true statement then \(p\vee t\) is a true statement.
In the language of logic: The disjunction of a true statement with any statement is a true statement.

 

[love/hatewithMath]

your text mean the tt\large t is a statement which is always true

That makes sense in the context of a truth table in which only true cases of t are shown. But unless I'm mistaken, it is possible for t to be false (as shown in my table); and the text of the problem doesn't state that t is always true.

 

[Dr.Peterson]

That makes sense in the context of a truth table in which only true cases of t are shown. But unless I'm mistaken, it is possible for t to be false (as shown in my table); and the text of the problem doesn't state that t is always true.

Somewhere, they must have told you what they mean by t. Look again. It may be a general notation they use, not just for this one problem. Or it may just be further back in the chapter.

What book is it? If I were with you in person, I would be looking through it. Maybe there is even an entry for "t" in the index.

 

[love/hatewithMath]

Look again. It may be a general notation they use, not just for this one problem. Or it may just be further back in the chapter.

I'll do that. The book is Epp's 4th ed. of Discrete Mathematics with Applications. Section 2.1

 

[love/hatewithMath]

Having scanned the entire section, I find no reference to either a variable or constant known as "t". I've attached the portion of the chapter which introduces the concept of logical equivalence. On Logical equivalence p.2, I find a truth table which resembles the abridged version found in the solution for problem 18 of the section exercises, which I posted a copy of at the beginning of the thread (but no explanation offered for the abridged table). The page containing problem 18 is shown in the final attachment.

 

[Dr.Peterson]

I'll do that. The book is Epp's 4th ed. of Discrete Mathematics with Applications. Section 2.1

Thanks. I happen to have the same book. (I haven't taught from it, but have tutored students who used it.)

You can see this t (bold) in Example 2.1.13, which is very much like your exercise; there they state, "If t is a tautology ..." They also use it in Theorem 2.1.11, which involves "a tautology t".

The truth table only needs to include the value T for t, because by definition that is the value of a tautology; it can never be F. This is what you were told in post #6.

It would have been better if the exercise had been worded like the example, since they don't seem to be using t as a general symbol for any tautology, but just as a locally defined entity within the example and the theorem; but they evidently expect you to recall it from reading the text, and to assume it is being used the same way.

 

[pka]

I'll do that. The book is Epp's 4th ed. of Discrete Mathematics with Applications. Section 2.1

Thanks. I happen to have the same book. (I haven't taught from it, but have tutored students who used it.

I do not have any of Susanna S Epp's books. One bad review put an end my reviews.
Here is one example. Can a tautology be false? In a table:
\(\begin{array}{*{20}{c}} P&t&{P \vee t} \\ \hline T&T&T \\ T&F&T \\ F&T&T \\ F&F&F \end{array}\)
That makes little sense if \(\bf t\) is a tautology.

 

[JeffM]

I do not have any of Susanna S Epp's books. One bad review put an end my reviews.
Here is one example. Can a tautology be false? In a table:
\(\begin{array}{*{20}{c}} P&t&{P \vee t} \\ \hline T&T&T \\ T&F&T \\ F&T&T \\ F&F&F \end{array}\)
That makes little sense if \(\bf t\) is a tautology.

You read one bad review of her work and therefore never read any other review, let alone anything she actually wrote?

 

[pka]

You read one bad review of her work and therefore never read any other review, let alone anything she actually wrote?

No, the publisher did not send any more free examination textbooks.

 

[JeffM]

No, the publisher did not send any more free examination textbooks.

Ahh. They want blurbs rather than reviews. Makes sense from a publisher's perspective.

So the professional journals don't independently BUY textbooks and assign a reviewer? Seems to me like a very useful function for them to take on.