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.

### Watch This

CK-12 Foundation: Chapter2TruthTablesA

### Guidance

So far we know these symbols for logic:

- not (negation)

- if-then

- therefore

Two more symbols are:

- and

- or

We would write “ and ” as and “ or ” as .

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

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

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 and .

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

Notice all the combinations of and . **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 is true, based upon the first two columns. **can only be true if BOTH and 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 and .

First, make columns for and , just like Example A.

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

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

#### Example C

Determine the truths for .

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.**

Next, address the . It will just be the opposites of the column.

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

Finally, we can address the entire problem, . Use the and to determine the values. Remember, for “and” both and must be true.

Watch this video for help with the Examples above.

CK-12 Foundation: Chapter2TruthTablesB

### Guided Practice

Write a truth table for the following variables.

1.

2.

3.

**Answers:**

1. First, make columns for , then add in and finally, evaluate .

2. First, make columns for and , then add in and . Finally, evaluate .

3. First, make columns for and , then add in and . Finally, evaluate .

### Explore More

Write a truth table for the following variables.

- The only difference between #1 and #3 is the placement of the parenthesis. How do the truth tables differ?
- When is true?

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

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 2.6.