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

[solomon_13000]

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?

 

[soroban]

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?

Have you never done one with three statements?

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

 

[solomon_13000]

so basically r is represented as true, false, true, false.....

 

[tegra97]

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

 

[soroban]

Hello, solomon_13000!

so basically r is represented as true, false, true, false.....

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- ...