3 Truth Tables and Analyzing Arguments: Examples
Truth Tables
Because complex Boolean statements can get tricky to think about, we can create a truth table to keep track of what truth values for the simple statements make the complex statement true and false
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 1
Suppose you’re picking out a new couch, and your significant other says “get a sectional or something with a chaise.”
This is a complex statement made of two simpler conditions: “is a sectional,” and “has a chaise.” For simplicity, let’s use S to designate “is a sectional,” and C to designate “has a chaise.” The condition S is true if the couch is a sectional.
A truth table for this would look like this:
| S | C | S or C |
|---|---|---|
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
In the table, T is used for true, and F for false. In the first row, if S is true and C is also true, then the complex statement “S or C” is true. This would be a sectional that also has a chaise, which meets our desire.
Remember also that or in logic is not exclusive; if the couch has both features, it does meet the condition.
To shorthand our notation further, we’re going to introduce some symbols that are commonly used for and, or, and not.
Symbols
The symbol ⋀ is used for and: A and B is notated A ⋀ B.
The symbol ⋁ is used for or: A or B is notated A ⋁ B
The symbol ~ is used for not: not A is notated ~A
You can remember the first two symbols by relating them to the shapes for the union and intersection. A ⋀ B would be the elements that exist in both sets, in A ⋂ B. Likewise, A ⋁ B would be the elements that exist in either set, in A ⋃ B.
In the previous example, the truth table was really just summarizing what we already know about how the or statement work. The truth tables for the basic and, or, and not statements are shown below.
Basic Truth Tables
| A | B | A ⋀ B |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
| A | B | A ⋁ B |
|---|---|---|
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
| A | ~A |
|---|---|
| T | F |
| F | T |
Truth tables really become useful when analyzing more complex Boolean statements.
Example 2
Create a truth table for the statement A ⋀ ~(B ⋁ C)
It helps to work from the inside out when creating truth tables, and create tables for intermediate operations. We start by listing all the possible truth value combinations for A, B, and C. Notice how the first column contains 4 Ts followed by 4 Fs, the second column contains 2 Ts, 2 Fs, then repeats, and the last column alternates. This pattern ensures that all combinations are considered. Along with those initial values, we’ll list the truth values for the innermost expression, B ⋁ C.
| A | B | C | B ⋁ C |
| T | T | T | T |
| T | T | F | T |
| T | F | T | T |
| T | F | F | F |
| F | T | T | T |
| F | T | F | T |
| F | F | T | T |
| F | F | F |