What Is the Correct Symbolic Form and Truth Table for ~(p|q)?

[SmokeyMTNJim]

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I am currently working through a Finite math book Intro to finite math: second Edition Kemeny, Snell, and Thompson. One of the exercises wants me to construct a truth table for the following:
~(p|q) earlier I am told that let p|q express that "p and q are not both true". Earlier I worked out that the symbolic form of this statement (p|q) to be ~(p/\~q).

my work on constructing a truth table for ~(p|q)
p|q ~ ~ (p /\ ~ q)
t t F T t f f t
t f T F t t t f
f t F T f f f t
f f F T f f t f

From here I thought I was to answer the ~ closest to (p, by countering what was under /\, giving me T F T T and then further negating that, ending with F T F F. This is wrong according to the book. as this truth table should end with T F F F.

2 questions: Is my symbolic form of ~(p|q) wrong and therefore my answer wrong, and/or, did I work something out wrong giving me the wrong answer. I think I probably made the symbolic version wrong but am not sure of how to go about this.

 

[haruspex]

This doesn't look right to me at all.
I have always understood p|q to represent "p or q". That isn't "p and q are not both true", which would be (~p)|(~q), or equivalently ~(p^q). It certainly cannot be ~(p/\~q) since that loses the symmetry between p and q.

 

[SmokeyMTNJim]

I can assure you that this book states :
7. let p|q express that "p and q are not both true." write a symbolic expression for p|q using ~ and /\. This textbook uses the notation \/ for or, /\ for and, ~ for negate.
I hope this helps and thank you for your response

 

[haruspex]

I can assure you that this book states :
7. let p|q express that "p and q are not both true." write a symbolic expression for p|q using ~ and /\. This textbook uses the notation \/ for or, /\ for and, ~ for negate.
I hope this helps and thank you for your response

Ok, that's fine, but that does not make it ~(p^~q). As I wrote, that is not symmetric in p and q. Let's get that right first.

 

[SmokeyMTNJim]

Okay, I do understand that what I did was incorrect. I am just trying to figure out how to solve this. I am having trouble tying to break it down into simple statements (p|q) = p and q are not both true. Simple statements: p is true, q is true. Here i don't know what to do about the both clause, because i don't think these two statements represent the given symbolic statement. I tried using exclusive disjunction ~ (p|q) = ~(p \/ q) but this truth table did not math what the answer in the book says. the book says the truth table should be TFFF.

Any insight on what I am doing wrong, or missing?

 

[haruspex]

(p|q) = p and q are not both true.

Right. If they are not both true then at least one of them is. ...?
Can you put that into English in the form "either ... or ..."?

 

[andrewkirk]

this book states :
7. let p|q express that "p and q are not both true."

I cannot understand the truth table in the OP because the column headings are all over the place.
However "p and q are not both true" means the same as "not (p and q are both true)", which is symbolised as $\neg(p\wedge q)$. So you can make a truth table by just reversing every entry in the standard truth table for the conjunction $p\wedge q$.

As I understand it, you have been asked to symbolise the negation of p|q, in other words $\neg(\neg(p\wedge q))$. If you are using Classical Logic, rather than some fancy version like Intuitionist or Minimal Logic, you can use 'double negation elimination' to simlify that.