# Construct a truth table for the compound proposition

• Aug 31st 2013, 12:43 PM
lamentofking
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.
• Aug 31st 2013, 12:57 PM
emakarov
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)
• Aug 31st 2013, 05:57 PM
Soroban
Re: Construct a truth table for the compound proposition
Hello, lamentofking!

Your question indicates that this is your first truth table.

Quote:

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

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

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