Construct a truth table for the compound proposition

I need to make a truth table for this compound proposition:

(q →¬p) ↔ (p ↔ q)

When looking at examples I noticed that between p and q, p always comes first in the truth table. So do I need to switch around the proposition to say:

(p ↔ q) ↔ (q →¬p) ?

I feel like if I leave it as is originally then instead of putting the table like this:

I would have to do this:

Am I over-thinking the problem or am I on the right track? Someone please provide some assistance. I am so lost.

Re: Construct a truth table for the compound proposition

Quote:

Originally Posted by

**lamentofking** Am I over-thinking the problem

Yes. In a truth table, variable names (i.e., names of columns) are usually listed in the alphabetical order, but this is not a law. In a formula, variables can occur in an arbitrary order. Formulas are just given to you, and you are not supposed to change them just to construct a truth table. There is nothing wrong in having a truth table with two columns p and q and the third column listing the truth values of the formula (q →¬p) ↔ (p ↔ q). This is just like having the following table listing values of an arithmetic expression.

Code:

`x y | y * x`

1 2 | 2

3 5 | 15

4 2 | 8

3 3 | 9

Here there is no need to switch the x and y columns even though the first variable in (y * x) is y.

You could also make the q column first. It would still be a valid truth table, but the order of truth values in the third column would be different, which may make it more difficult for your instructor to check your work. (Angry)

Re: Construct a truth table for the compound proposition

Hello, lamentofking!

Your question indicates that this is your *first* truth table.

Quote:

$\displaystyle \text{Construct a truth table for: }\:(q \to\,\sim\!p)\:\longleftrightarrow\:(p \leftrightarrow q)$

$\displaystyle \begin{array}{|c|c|ccccccc|} p & q & (q & \to & \sim\!p) & \longleftrightarrow & (p & \leftrightarrow & q) \\ \hline T&T& T&F&F & \color{blue}{F} &T&T&T \\ T&F & F&T&F & \color{blue}{F} & T&F&F \\ F&T & T&T&T & \color{blue}{F} & F&F&T \\ F&F & F&T&T & \color{blue}{T} & F&T&F \\ \hline &&1&2&1 &3& 1&2&1 \\ \hline\end{array}$

The numbers indicate the order in which the columns are filled.