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Truth Tables

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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:

  • \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.

CK-12 Foundation: Chapter2TruthTablesB

Vocabulary

Truth tables use symbols to analyze logic.

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)

Answers:

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

Practice

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

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