Constructing Truth table for (p \/ q) /\ r

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solomon_13000

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Construct a truth table for each of these compound propositions:

(p \/ q) /\ r

The part I am confuse is r. How do I know the value for r using the truth table?
 
Re: Truth table

Hello, solomon_13000!

Construct a truth table for each of these compound propositions:

. . \(\displaystyle (p\,\vee\,q)\,\wedge\,r\)

The part I am confuse is r.
How do I know the value for r using the truth table?
Click to expand...

Have you never done one with three statements?

. . \(\displaystyle \begin{array}{ccccc} p & | & q & | & r \\
\hline \\T & | & T & | & T \\T & | & T & | & F \\T & | & F & | & T \\T & | & F & | & F \\
F & | & T & | & T \\F & | & T & | & F \\F & | & F & | & T \\F & | & F & | & F\end{array}\)

 
You want get all the possible true and false combinations of p,q, and r.
Your table should look like this:
p | q | r | pVq | (pVq) /\ r
 
Hello, solomon_13000!

so basically r is represented as true, false, true, false.....
Click to expand...

Let me clarify this . . .

With \(\displaystyle n\) statements, there will be \(\displaystyle 2^n\) cases.


Suppose we have four statements: \(\displaystyle p,\,q,\,r,\,s\)

There will be \(\displaystyle 2^4\,=\,16\) cases.

How can we make sure we have all of them?
. . Well, there is a pattern to the T's and F's.

The truth table looks like this:

. . \(\displaystyle \begin{array}{ccccccc} p & | & q & | & r & | & s\\
\hline \\
T & | & T & | & T & | & T \\
T & | & T & | & T & | & F \\
T & | & T & | & F & | & T \\
T & | & T & | & F & | & F \\
T & | & F & | & T & | & T \\
T & | & F & | & T & | & F \\
T & | & F & | & F & | & T \\
T & | & F & | & F & | & F \\
F & | & T & | & T & | & T \\
F & | & T & | & T & | & F \\
F & | & T & | & F & | & T \\
F & | & T & | & F & | & F \\
F & | & F & | & T & | & T \\
F & | & F & | & T & | & F \\
F & | & F & | & F & | & T \\
F & | & F & | & F & | & F\end{array}\)

In the \(\displaystyle p\) column, there are 8 T's and 8 F's.
. . The first half is T's; the second half is F's.

In the \(\displaystyle q\) coluimn, there are 4 T's, 4 F's, 4 T's, 4 F's.
. . They "change every four".

In the \(\displaystyle r\) column, they "change every two".

In the last column, they alternate: T-F-T-F- ...

 
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