# Math Help - Constructing a truth table using the connective "not"

1. ## Constructing a truth table using the connective "not"

I am trying to construct a truth table for the statements 'P' and 'Q' - I am not told what these statements are so I guess they're random, anything ones, and I now need to construct a truth table for the statement:

(not P) or (not Q):

I did:

P...Q.....(not P)...(not Q).....(not P) or (not Q)

T...T........F...........F....................T
T...F........F...........T...................
F...T........T...........F..................
F...F........T...........T

That was my attempt at drawing a table, the very top line is the heading bit of the table, the rest are its "entries", which hopefully are in the right spaces, if you want me to clear it up, please ask

I am not sure what to put into the final (not P) or (not Q) column. Does this column mean that you want either (not P) to be true OR (not Q) to be true, as P and Q are both true, from the first 2 columns?

I am not too sure what I am looking for to put into the final column?

If you want to know the exact question, it is from Peter J. Eccles's "An introduction to mathematical reasoning", in chapter one, exercises 1.2.

Thank you

2. Hello, ironz!

It sounds like this is your first truth table . . . is it?

$\text{Construct a truth table for the statement: }\; \sim p \:\vee \sim q$

You are off to a good start.

. . $\begin{array}{|c|c||c|c||x|x}
p & q & \sim p & \sim q & \sim p\:\vee\sim q \\ \hline
T & T & F & F & \\ T & F & F & T & \\ F & T & T & F & \\ F & F & T & T & \\ \hline
_1 & _2 & _3 & _4 & _5 \\ \hline \end{array}$

Now you want an "or" between columns 3 and 4.

You're expected to when an "or" (disjunction) is true.

. . . . . $\begin{array}{cccc}T \vee T &=& T \\ T \vee F &=& T \\ F \vee T &=& T \\ F \vee F &=& F \end{array}$

Now you can complete the fifth column:

. . $\begin{array}{|c|c||c|c||x|x}
p & q & \sim p & \sim q & \sim p\:\vee\sim q \\ \hline
T & T & F & F & {\bf F} \\ T & F & F & T & {\bf T} \\ F & T & T & F & {\bf T} \\ F & F & T & T & {\bf T} \\ \hline
_1 & _2 & _3 & _4 & _5 \\ \hline \end{array}$

3. Originally Posted by Soroban
Hello, ironz!

It sounds like this is your first truth table . . . is it?

You are off to a good start.

. . $\begin{array}{|c|c||c|c||x|x}
p & q & \sim p & \sim q & \sim p\:\vee\sim q \\ \hline
T & T & F & F & \\ T & F & F & T & \\ F & T & T & F & \\ F & F & T & T & \\ \hline
_1 & _2 & _3 & _4 & _5 \\ \hline \end{array}$

Now you want an "or" between columns 3 and 4.

You're expected to when an "or" (disjunction) is true.

. . . . . $\begin{array}{cccc}T \vee T &=& T \\ T \vee F &=& T \\ F \vee T &=& T \\ F \vee F &=& F \end{array}$

Now you can complete the fifth column:

. . $\begin{array}{|c|c||c|c||x|x}
p & q & \sim p & \sim q & \sim p\:\vee\sim q \\ \hline
T & T & F & F & {\bf F} \\ T & F & F & T & {\bf T} \\ F & T & T & F & {\bf T} \\ F & F & T & T & {\bf T} \\ \hline
_1 & _2 & _3 & _4 & _5 \\ \hline \end{array}$

Yes, this is my first one lol.

I still don't understand how you got the answers for the final column. What are you looking for? Is it because the bottom 3 have a TRUE in them and as it is either one OR the other, you only need one TRUE to make the whole statement true. But the first one has both of them as FALSE and so there are no TRUE statements here.

4. Originally Posted by ironz

I still don't understand how you got the answers for the final column. What are you looking for? Is it because the bottom 3 have a TRUE in them and as it is either one OR the other, you only need one TRUE to make the whole statement true. But the first one has both of them as FALSE and so there are no TRUE statements here.
You got it. You want at least one of $\neg p$ or $\neg q$ to be true. The 'not' can be confusing, but it is a proposition just like p.
Also, in boolean logic 'or' is inclusive, meaning $p\vee q$ is true if p, q, or both p and q are true.