Learning Objectives
- What is a Truth Table?
- Basic Truth Tables for
- Negation
- Conjunction
- Disjunction
- Order of Logical Operations
- Putting together (negation, conjunction, disjunction)
Because complex statements can get tricky to think about, we can create a truth table to break the complex statement into simple statements, and determine whether they are true or false. A table will help keep track of all the truth values of the simple statements that make up a complex statement, leading to an analysis of the full statement.
Truth Table
A table showing what the resulting truth value of a complex statement is for all the possible truth values for the simple statements.
Example
Suppose you’re picking out a new couch, and your significant other says “get a sectional or something with a chaise.” Construct a truth table that describes the elements of the conditions of this statement and whether the conditions are met.
Negation tells us, “It is not the case that… ”
Conjunction tells us, “Both… are the case.” Conjunctions are only true when both conjuncts are true.
Disjunction tells us that, “At least one is the case… ” Disjunctions are only false when both disjuncts are false.
Basic Truth Tables
Notice that the b column has the pattern TFTF. Then the A column has the pattern TTFF.
| a | b | [latex]a\wedge{b}[/latex] |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
| a | b | [latex]a\vee{b}[/latex] |
|---|---|---|
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
When thinking of “or” situations, it is also helpful to think of a scenario. Imagine you are a renter (like in the previous example) and your landlord asked you to pay on the 1st of pay on the 2nd. Suppose A is pay on the first and B is pay on the second. The result will be true in each scenario except when you don’t pay on the 1st or on the second. In that case the result is false.
Truth tables for “or” use the “inclusive or“, meaning A or B or both. For example: You can A (bring your umbrella) or B (wear a rain coat) or both (bring your umbrella and wear a rain coat). “Exclusive or“, on the other hand, means A or B, but not both. For example, you can A (take your lottery winnings in one lump sum) or B (take you lottery winnings as monthly payments), but you cannot do both.
| a | [latex]\sim{a}[/latex] |
|---|---|
| T | F |
| F | T |
Note:
When we create the truth table, we need to list all the possible truth value combinations for A and B. Notice how the first column contains 2 True’s (T) followed by 2 False’s (F). The second column alternates T, F, T, F. This pattern ensures that all 4 combinations are considered.
| A | B |
| T | T |
| T | F |
| F | T |
| F | F |
Try It
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