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Truth Tables

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Truth Tables

What if you needed to analyze a complex logical argument? How could you do this is an organized way, making sure to account for everything? After completing this Concept, you'll be able to use truth tables as a way to organize and analyze logic.

Guidance

So far we know these symbols for logic:

• $\sim$ not (negation)
• $\rightarrow$ if-then
• $\therefore$ therefore

Two more symbols are:

• $\land$ and
• $\lor$ or

We would write “ $p$ and $q$ ” as $p \land q$ and “ $p$ or $q$ ” as $p \lor q$ .

Truth tables use these symbols and are another way to analyze logic. First, let’s relate $p$ and $\sim p$ . To make it easier, set $p$ as: An even number. Therefore, $\sim p$ is An odd number . Make a truth table to find out if they are both true. Begin with all the “truths” of $p$ , true (T) or false (F).

$p$
$T$
$F$

Next we write the corresponding truth values for $\sim p$ . $\sim p$ has the opposite truth values of $p$ . So, if $p$ is true, then $\sim p$ is false and vise versa.

$p$ $\sim p$
T F
F T

To Recap:

• Start truth tables with all the possible combinations of truths. For 2 variables there are 4 combinations for 3 variables there are 8. You always start a truth table this way.
• Do any negations on the any of the variables.
• Do any combinations in parenthesis.
• Finish with completing what the problem was asking for.

Example A

Draw a truth table for $p, q$ and $p \land q$ .

First, make columns for $p$ and $q$ . Fill the columns with all the possible true and false combinations for the two.

$p$ $q$
$T$ $T$
$T$ $F$
$F$ $T$
$F$ $F$

Notice all the combinations of $p$ and $q$ . Anytime we have truth tables with two variables, this is always how we fill out the first two columns.

Next, we need to figure out when $p \land q$ is true, based upon the first two columns. $p \land q$ can only be true if BOTH $p$ and $q$ are true. So, the completed table looks like this:

This is how a truth table with two variables and their “and” column is always filled out.

Example B

Draw a truth table for $p, q$ and $p \lor q$ .

First, make columns for $p$ and $q$ , just like Example A.

$p$ $q$
$T$ $T$
$T$ $F$
$F$ $T$
$F$ $F$

Next, we need to figure out when $p \lor q$ is true, based upon the first two columns. $p \lor q$ is true if $p$ OR $q$ are true, or both are true. So, the completed table looks like this:

The difference between $p \land q$ and $p \lor q$ is the second and third rows. For “and” both $p$ and $q$ have to be true, but for “or” only one has to be true.

Example C

Determine the truths for $p \land (\sim q \lor r)$ .

First, there are three variables, so we are going to need all the combinations of their truths. For three variables, there are always 8 possible combinations.

$p$ $q$ $r$
$T$ $T$ $T$
$T$ $T$ $F$
$T$ $F$ $T$
$T$ $F$ $F$
$F$ $T$ $T$
$F$ $T$ $F$
$F$ $F$ $T$
$F$ $F$ $F$

Next, address the $\sim q$ . It will just be the opposites of the $q$ column.

$p$ $q$ $r$ $\sim q$
$T$ $T$ $T$ $F$
$T$ $T$ $F$ $F$
$T$ $F$ $T$ $T$
$T$ $F$ $F$ $T$
$F$ $T$ $T$ $F$
$F$ $T$ $F$ $F$
$F$ $F$ $T$ $T$
$F$ $F$ $F$ $T$

Now, let’s do what’s in the parenthesis, $\sim q \lor r$ . Remember, for “or” only $\sim q$ OR $r$ has to be true. Only use the $\sim q$ and $r$ columns to determine the values in this column.

$p$ $q$ $r$ $\sim q$ $\sim q \lor r$
$T$ $T$ $T$ $F$ $T$
$T$ $T$ $F$ $F$ $F$
$T$ $F$ $T$ $T$ $T$
$T$ $F$ $F$ $T$ $T$
$F$ $T$ $T$ $F$ $T$
$F$ $T$ $F$ $F$ $F$
$F$ $F$ $T$ $T$ $T$
$F$ $F$ $F$ $T$ $T$

Finally, we can address the entire problem, $p \land (\sim q \lor r)$ . Use the $p$ and $\sim q \lor r$ to determine the values. Remember, for “and” both $p$ and $\sim q \lor r$ must be true.

$p$ $q$ $r$ $\sim q$ $\sim q \lor r$ $p \land(\sim q \lor r)$
$T$ $T$ $T$ $F$ $T$ $T$
$T$ $T$ $F$ $F$ $F$ $F$
$T$ $F$ $T$ $T$ $T$ $T$
$T$ $F$ $F$ $T$ $T$ $T$
$F$ $T$ $T$ $F$ $T$ $F$
$F$ $T$ $F$ $F$ $F$ $F$
$F$ $F$ $T$ $T$ $T$ $F$
$F$ $F$ $F$ $T$ $T$ $F$

Watch this video for help with the Examples above.

Guided Practice

Write a truth table for the following variables.

1. $p \land \sim p$

2. $\sim p \lor \sim q$

3. $p \land (q \lor \sim q)$

1. First, make columns for $p$ , then add in $\sim p$ and finally, evaluate $p \land \sim p$ .

$p$ $\sim p$ $p\land \sim p$
$T$ $F$ $F$
$F$ $T$ $F$

2. First, make columns for $p$ and $q$ , then add in $\sim p$ and $\sim q$ . Finally, evaluate $\sim p \lor \sim q$ .

$p$ $q$ $\sim p$ $\sim q$ $\sim p \lor \sim q$
$T$ $T$ $F$ $F$ $F$
$T$ $F$ $F$ $T$ $T$
$F$ $T$ $T$ $F$ $T$
$F$ $F$ $T$ $T$ $T$

3. First, make columns for $p$ and $q$ , then add in $\sim q$ and $q \lor \sim q$ . Finally, evaluate $p \land (q \lor \sim q)$ .

$p$ $q$ $\sim q$ $(q \lor \sim q)$ $p \land (q \lor \sim q)$
$T$ $T$ $F$ $T$ $T$
$T$ $F$ $T$ $T$ $T$
$F$ $T$ $F$ $T$ $F$
$F$ $F$ $T$ $T$ $F$

Explore More

Write a truth table for the following variables.

1. $(p \land q) \lor \sim r$
2. $p \lor (\sim q \lor r)$
3. $p \land (q \lor \sim r)$
4. The only difference between #1 and #3 is the placement of the parenthesis. How do the truth tables differ?
5. When is $p \lor q \lor r$ true?
6. $p \lor q \lor r$
7. $(p \lor q) \lor \sim r$
8. $(\sim p \land \sim q) \land r$
9. $(\sim p \lor \sim q) \land r$

Is the following a valid argument? If so, what law is being used? HINT: Statements could be out of order.

1. $p \rightarrow q \\r \rightarrow p\\\therefore r \rightarrow q$
2. $p \rightarrow q\\r \rightarrow q\\\therefore p \rightarrow r$
3. $p \rightarrow \sim r\\r\\\therefore \sim p$
4. $\sim q \rightarrow r\\q\\\therefore \sim r$
5. $p \rightarrow (r \rightarrow s)\\p\\\therefore r \rightarrow s$
6. $r \rightarrow q\\r \rightarrow s\\\therefore q \rightarrow s$