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Truth Tables

Organization and analysis of logic using symbols.

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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.

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CK-12 Foundation: Chapter2TruthTablesA

James Sousa: Truth Tables

Guidance

So far we know these symbols for logic:

  • \begin{align*}\sim\end{align*} not (negation)
  • \begin{align*}\rightarrow\end{align*} if-then
  • \begin{align*}\therefore\end{align*} therefore

Two more symbols are:

  • \begin{align*}\land\end{align*} and
  • \begin{align*}\lor\end{align*} or

We would write “\begin{align*}p\end{align*} and \begin{align*}q\end{align*}” as \begin{align*}p \land q\end{align*} and “\begin{align*}p\end{align*} or \begin{align*}q\end{align*}” as \begin{align*}p \lor q\end{align*}.

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

\begin{align*}p\end{align*}
\begin{align*}T\end{align*}
\begin{align*}F\end{align*}

Next we write the corresponding truth values for \begin{align*}\sim p\end{align*}. \begin{align*}\sim p\end{align*} has the opposite truth values of \begin{align*}p\end{align*}. So, if \begin{align*}p\end{align*} is true, then \begin{align*}\sim p\end{align*} is false and vise versa.

\begin{align*}p\end{align*} \begin{align*}\sim p\end{align*}
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 \begin{align*}p, q\end{align*} and \begin{align*}p \land q\end{align*}.

First, make columns for \begin{align*}p\end{align*} and \begin{align*}q\end{align*}. Fill the columns with all the possible true and false combinations for the two.

\begin{align*}p\end{align*} \begin{align*}q\end{align*}
\begin{align*}T\end{align*} \begin{align*}T\end{align*}
\begin{align*}T\end{align*} \begin{align*}F\end{align*}
\begin{align*}F\end{align*} \begin{align*}T\end{align*}
\begin{align*}F\end{align*} \begin{align*}F\end{align*}

Notice all the combinations of \begin{align*}p\end{align*} and \begin{align*}q\end{align*}. 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 \begin{align*}p \land q\end{align*} is true, based upon the first two columns. \begin{align*}p \land q\end{align*} can only be true if BOTH \begin{align*}p\end{align*} and \begin{align*}q\end{align*} 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 \begin{align*}p, q\end{align*} and \begin{align*}p \lor q\end{align*}.

First, make columns for \begin{align*}p\end{align*} and \begin{align*}q\end{align*}, just like Example A.

\begin{align*}p\end{align*} \begin{align*}q\end{align*}
\begin{align*}T\end{align*} \begin{align*}T\end{align*}
\begin{align*}T\end{align*} \begin{align*}F\end{align*}
\begin{align*}F\end{align*} \begin{align*}T\end{align*}
\begin{align*}F\end{align*} \begin{align*}F\end{align*}

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

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

Example C

Determine the truths for \begin{align*}p \land (\sim q \lor r)\end{align*}.

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.

\begin{align*}p\end{align*} \begin{align*}q\end{align*} \begin{align*}r\end{align*}
\begin{align*}T\end{align*} \begin{align*}T\end{align*} \begin{align*}T\end{align*}
\begin{align*}T\end{align*} \begin{align*}T\end{align*} \begin{align*}F\end{align*}
\begin{align*}T\end{align*} \begin{align*}F\end{align*} \begin{align*}T\end{align*}
\begin{align*}T\end{align*} \begin{align*}F\end{align*} \begin{align*}F\end{align*}
\begin{align*}F\end{align*} \begin{align*}T\end{align*} \begin{align*}T\end{align*}
\begin{align*}F\end{align*} \begin{align*}T\end{align*} \begin{align*}F\end{align*}
\begin{align*}F\end{align*} \begin{align*}F\end{align*} \begin{align*}T\end{align*}
\begin{align*}F\end{align*} \begin{align*}F\end{align*} \begin{align*}F\end{align*}

Next, address the \begin{align*}\sim q\end{align*}. It will just be the opposites of the \begin{align*}q\end{align*} column.

\begin{align*}p\end{align*} \begin{align*}q\end{align*} \begin{align*}r\end{align*} \begin{align*}\sim q\end{align*}
\begin{align*}T\end{align*} \begin{align*}T\end{align*} \begin{align*}T\end{align*} \begin{align*}F\end{align*}
\begin{align*}T\end{align*} \begin{align*}T\end{align*} \begin{align*}F\end{align*} \begin{align*}F\end{align*}
\begin{align*}T\end{align*} \begin{align*}F\end{align*} \begin{align*}T\end{align*} \begin{align*}T\end{align*}
\begin{align*}T\end{align*} \begin{align*}F\end{align*} \begin{align*}F\end{align*} \begin{align*}T\end{align*}
\begin{align*}F\end{align*} \begin{align*}T\end{align*} \begin{align*}T\end{align*} \begin{align*}F\end{align*}
\begin{align*}F\end{align*} \begin{align*}T\end{align*} \begin{align*}F\end{align*} \begin{align*}F\end{align*}
\begin{align*}F\end{align*} \begin{align*}F\end{align*} \begin{align*}T\end{align*} \begin{align*}T\end{align*}
\begin{align*}F\end{align*} \begin{align*}F\end{align*} \begin{align*}F\end{align*} \begin{align*}T\end{align*}

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

\begin{align*}p\end{align*} \begin{align*}q\end{align*} \begin{align*}r\end{align*} \begin{align*}\sim q\end{align*} \begin{align*}\sim q \lor r\end{align*}
\begin{align*}T\end{align*} \begin{align*}T\end{align*} \begin{align*}T\end{align*} \begin{align*}F\end{align*} \begin{align*}T\end{align*}
\begin{align*}T\end{align*} \begin{align*}T\end{align*} \begin{align*}F\end{align*} \begin{align*}F\end{align*} \begin{align*}F\end{align*}
\begin{align*}T\end{align*} \begin{align*}F\end{align*} \begin{align*}T\end{align*} \begin{align*}T\end{align*} \begin{align*}T\end{align*}
\begin{align*}T\end{align*} \begin{align*}F\end{align*} \begin{align*}F\end{align*} \begin{align*}T\end{align*} \begin{align*}T\end{align*}
\begin{align*}F\end{align*} \begin{align*}T\end{align*} \begin{align*}T\end{align*} \begin{align*}F\end{align*} \begin{align*}T\end{align*}
\begin{align*}F\end{align*} \begin{align*}T\end{align*} \begin{align*}F\end{align*} \begin{align*}F\end{align*} \begin{align*}F\end{align*}
\begin{align*}F\end{align*} \begin{align*}F\end{align*} \begin{align*}T\end{align*} \begin{align*}T\end{align*} \begin{align*}T\end{align*}
\begin{align*}F\end{align*} \begin{align*}F\end{align*} \begin{align*}F\end{align*} \begin{align*}T\end{align*} \begin{align*}T\end{align*}

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

\begin{align*}p\end{align*} \begin{align*}q\end{align*} \begin{align*}r\end{align*} \begin{align*}\sim q\end{align*} \begin{align*}\sim q \lor r\end{align*} \begin{align*}p \land(\sim q \lor r)\end{align*}
\begin{align*}T\end{align*} \begin{align*}T\end{align*} \begin{align*}T\end{align*} \begin{align*}F\end{align*} \begin{align*}T\end{align*} \begin{align*}T\end{align*}
\begin{align*}T\end{align*} \begin{align*}T\end{align*} \begin{align*}F\end{align*} \begin{align*}F\end{align*} \begin{align*}F\end{align*} \begin{align*}F\end{align*}
\begin{align*}T\end{align*} \begin{align*}F\end{align*} \begin{align*}T\end{align*} \begin{align*}T\end{align*} \begin{align*}T\end{align*} \begin{align*}T\end{align*}
\begin{align*}T\end{align*} \begin{align*}F\end{align*} \begin{align*}F\end{align*} \begin{align*}T\end{align*} \begin{align*}T\end{align*} \begin{align*}T\end{align*}
\begin{align*}F\end{align*} \begin{align*}T\end{align*} \begin{align*}T\end{align*} \begin{align*}F\end{align*} \begin{align*}T\end{align*} \begin{align*}F\end{align*}
\begin{align*}F\end{align*} \begin{align*}T\end{align*} \begin{align*}F\end{align*} \begin{align*}F\end{align*} \begin{align*}F\end{align*} \begin{align*}F\end{align*}
\begin{align*}F\end{align*} \begin{align*}F\end{align*} \begin{align*}T\end{align*} \begin{align*}T\end{align*} \begin{align*}T\end{align*} \begin{align*}F\end{align*}
\begin{align*}F\end{align*} \begin{align*}F\end{align*} \begin{align*}F\end{align*} \begin{align*}T\end{align*} \begin{align*}T\end{align*} \begin{align*}F\end{align*}

Watch this video for help with the Examples above.

CK-12 Foundation: Chapter2TruthTablesB

Vocabulary

Truth tables use symbols to analyze logic.

Guided Practice

Write a truth table for the following variables.

1. \begin{align*}p \land \sim p\end{align*}

2. \begin{align*}\sim p \lor \sim q\end{align*}

3. \begin{align*}p \land (q \lor \sim q)\end{align*}

Answers:

1. First, make columns for \begin{align*}p\end{align*}, then add in \begin{align*}\sim p\end{align*} and finally, evaluate \begin{align*} p \land \sim p\end{align*}.

\begin{align*}p\end{align*} \begin{align*}\sim p\end{align*} \begin{align*}p\land \sim p\end{align*}
\begin{align*}T\end{align*} \begin{align*}F\end{align*} \begin{align*}F\end{align*}
\begin{align*}F\end{align*} \begin{align*}T\end{align*} \begin{align*}F\end{align*}

2. First, make columns for \begin{align*}p\end{align*} and \begin{align*}q\end{align*}, then add in \begin{align*}\sim p\end{align*} and \begin{align*}\sim q\end{align*}. Finally, evaluate \begin{align*}\sim p \lor \sim q\end{align*}.

\begin{align*}p\end{align*} \begin{align*}q\end{align*} \begin{align*}\sim p\end{align*} \begin{align*}\sim q\end{align*} \begin{align*}\sim p \lor \sim q\end{align*}
\begin{align*}T\end{align*} \begin{align*}T\end{align*} \begin{align*}F\end{align*} \begin{align*}F\end{align*} \begin{align*}F\end{align*}
\begin{align*}T\end{align*} \begin{align*}F\end{align*} \begin{align*}F\end{align*} \begin{align*}T\end{align*} \begin{align*}T\end{align*}
\begin{align*}F\end{align*} \begin{align*}T\end{align*} \begin{align*}T\end{align*} \begin{align*}F\end{align*} \begin{align*}T\end{align*}
\begin{align*}F\end{align*} \begin{align*}F\end{align*} \begin{align*}T\end{align*} \begin{align*}T\end{align*} \begin{align*}T\end{align*}

3. First, make columns for \begin{align*}p\end{align*} and \begin{align*}q\end{align*}, then add in \begin{align*}\sim q\end{align*} and \begin{align*}q \lor \sim q\end{align*}. Finally, evaluate \begin{align*}p \land (q \lor \sim q)\end{align*}.

\begin{align*}p\end{align*} \begin{align*}q\end{align*} \begin{align*}\sim q\end{align*} \begin{align*}(q \lor \sim q)\end{align*} \begin{align*}p \land (q \lor \sim q)\end{align*}
\begin{align*}T\end{align*} \begin{align*}T\end{align*} \begin{align*}F\end{align*} \begin{align*}T\end{align*} \begin{align*}T\end{align*}
\begin{align*}T\end{align*} \begin{align*}F\end{align*} \begin{align*}T\end{align*} \begin{align*}T\end{align*} \begin{align*}T\end{align*}
\begin{align*}F\end{align*} \begin{align*}T\end{align*} \begin{align*}F\end{align*} \begin{align*}T\end{align*} \begin{align*}F\end{align*}
\begin{align*}F\end{align*} \begin{align*}F\end{align*} \begin{align*}T\end{align*} \begin{align*}T\end{align*} \begin{align*}F\end{align*}

Practice

Write a truth table for the following variables.

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

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

  1. \begin{align*}p \rightarrow q \\ r \rightarrow p\\ \therefore r \rightarrow q\end{align*}
  2. \begin{align*}p \rightarrow q\\ r \rightarrow q\\ \therefore p \rightarrow r\end{align*}
  3. \begin{align*}p \rightarrow \sim r\\ r\\ \therefore \sim p\end{align*}
  4. \begin{align*}\sim q \rightarrow r\\ q\\ \therefore \sim r\end{align*}
  5. \begin{align*}p \rightarrow (r \rightarrow s)\\ p\\ \therefore r \rightarrow s\end{align*}
  6. \begin{align*}r \rightarrow q\\ r \rightarrow s\\ \therefore q \rightarrow s\end{align*}

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