[TechSyndrome]

Hi, I've been given a question in my Propositional Logic tutorial where I have to construct a truth table for this expression: (P ∨ ¬Q) → ¬(Q ∨ ¬P). Here's what I've managed so far:



1. I first drew up a truth table for the variables P and Q
2. I then separated the two given expressions by the '→' symbol
3. In the expression 'P ∨ ¬Q', '¬Q' is the inverse (opposite) of the truth value in the Q variable column, so I think my answers are correct in this column.

Now I've got a problem. I'm unsure as to what I should do in the '→' column. I tried applying this simple rule of implication - the → operation returns false, if and only if the first term is true, and the second term is false. So my answer for the '→' column was this:



But according to my solutions, this answer is wrong. Can anyone point me in the right direction?

p.s For some reason, my images are being rotated when I insert them. You might have to save them and rotate them (or tilt your head) to see it properly.

[r.stiltskin]

At the beginning of your post you wrote ¬(Q ∨ ¬P), but in your truth table you wrote ¬Q ∨ ¬P. Clearly (?) those are not the same.

So why don't you fill in values for (Q ∨ ¬P) as a column in your truth table? And then ¬(Q ∨ ¬P) as the next column.

[TechSyndrome]

r.stiltskin, on 11 June 2012 - 03:37 PM, said:

At the beginning of your post you wrote ¬(Q ∨ ¬P), but in your truth table you wrote ¬Q ∨ ¬P. Clearly (?) those are not the same.

So why don't you fill in values for (Q ∨ ¬P) as a column in your truth table? And then ¬(Q ∨ ¬P) as the next column.

Thank you for pointing that out. Just incase anyone else comes across this problem, I'll explain what I did.

1. I drew the truth table for variables P and Q.
2. I completed the truth table for the first expression P ∨ ¬Q
3. I completed the truth table for the second expression ¬(Q ∨ ¬P)
4. Then I completed the the truth table for the middle expression →

I thought I had to complete it in a sequence (one by one in order), but that's not the case. Variables, Expressions, then auxiliary symbols. Does that sound correct?

[TechSyndrome]

TechSyndrome, on 11 June 2012 - 03:48 PM, said:

r.stiltskin, on 11 June 2012 - 03:37 PM, said:

At the beginning of your post you wrote ¬(Q ∨ ¬P), but in your truth table you wrote ¬Q ∨ ¬P. Clearly (?) those are not the same.

So why don't you fill in values for (Q ∨ ¬P) as a column in your truth table? And then ¬(Q ∨ ¬P) as the next column.

Thank you for pointing that out. Just incase anyone else comes across this problem, I'll explain what I did.

1. I drew the truth table for variables P and Q.
2. I completed the truth table for the first expression P ∨ ¬Q
3. I completed the truth table for the second expression ¬(Q ∨ ¬P)
4. Then I completed the the truth table for the middle expression →

I thought I had to complete it in a sequence (one by one in order), but that's not the case. Variables, Expressions, then auxiliary symbols. Does that sound correct?

Oh, I should add that to get the result of ¬(Q ∨ ¬P) I had to get the result of the expression in the parentheses first i.e. Q ∨ ¬P. When you get that result, make sure you write the opposite truth value because that's what the ¬ does to it. If that's not clear to anyone, reply back and I'll do my best to explain what I did.

[TechSyndrome]

My lecturers explanation of the above ^:

"Neither a tautology (because principal column contains a 0) nor a contradiction (because principal column contains a 1)"

He wrote that it's neither a tautology nor a contradiction because the principle column has a 1 and a 0. I know what tautology is, and I know what contradiction is. But I don't know what he means by principle column. When I looked it up on YouTube, the chap who was explaining it said that the he looked at the last column to give him the answer. But he didn't explicitly use the expression principle column - is this just another way of saying "last column"?

This post has been edited by TechSyndrome: 11 June 2012 - 04:21 PM

[darek9576]

Never came across principle column. I would just use De Morgan law there.

[r.stiltskin]

TechSyndrome, on 11 June 2012 - 06:48 PM, said:

... Variables, Expressions, then auxiliary symbols. Does that sound correct?

I would shorten that to just variables and then expressions.

What do you mean by "auxiliary symbols"? Symbols are either variables or operators (connectives) which combine variables to form expressions and join expresssions to form more complex expressions. Your last step in this exercise wasn't to evaluate the truth value of the symbol →. An operator itself has no inherent truth value. Only an expression has a truth value (regarding a variable as a simple expression). What you were doing was assigning a truth value to the expression (P ∨ ¬Q) → ¬(Q ∨ ¬P) for each set of values of P and Q. Your "→" column is actually the "(P ∨ ¬Q) → ¬(Q ∨ ¬P)" column.

I've never come across the term "principle column" either. My guess is that your lecturer meant principal in the general sense of that word, e.g., main or most important, referring to the column representing the values of the "main" expression in your table.

[Dormilich]

r.stiltskin, on 12 June 2012 - 04:36 AM, said:

I've never come across the term "principle column" either. My guess is that your lecturer meant principal in the general sense of that word,

"principle column" is a typo/misreading of the OP. the lecturer indeed said "principal column" (see quote)