# Math Help - Construct a truth table for the compound proposition

1. ## 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:

 p q T T T F F T F F

I would have to do this:

 q p T T T F F T F F

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

2. ## Re: Construct a truth table for the compound proposition

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.

3. ## Re: Construct a truth table for the compound proposition

Hello, lamentofking!

$\text{Construct a truth table for: }\:(q \to\,\sim\!p)\:\longleftrightarrow\:(p \leftrightarrow q)$
$\begin{array}{|c|c|ccccccc|} p & q & (q & \to & \sim\!p) & \longleftrightarrow & (p & \leftrightarrow & q) \\ \hline T&T& T&F&F & \color{blue}{F} &T&T&T \\ T&F & F&T&F & \color{blue}{F} & T&F&F \\ F&T & T&T&T & \color{blue}{F} & F&F&T \\ F&F & F&T&T & \color{blue}{T} & F&T&F \\ \hline &&1&2&1 &3& 1&2&1 \\ \hline\end{array}$