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
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.