Chapter 7 Logic
7.3 Constructing Truth Tables
Learning Objectives
By the end of this section, you will be able to:
- Interpret and apply negations, conjunctions, and disjunctions
- Construct a truth table using negations, conjunctions, and disjunctions
- Construct a truth table for a compound statement and interpret its validity
Are you familiar with the Choose Your Own Adventure book series written by Edward Packard? These gamebooks allow the reader to become one of the characters and make decisions that affect what happens next, resulting in different sequences of events in the story and endings based on the choices made. Writing a computer program is a little like what it must be like to write one of these books. The programmer must consider all the possible inputs that a user can put into the program and decide what will happen in each case, then write their program to account for each of these possible outcomes.
A truth table is a graphical tool used to analyze all the possible truth values of the component logical statements to determine the validity of a statement or argument along with all its possible outcomes. The rows of the table correspond to each possible outcome for the given logical statement identified at the top of each column. A single logical statement [latex]p[/latex] has two possible truth values, true or false. In truth tables, a capital T will represent true values, and a capital F will represent false values.
In this section, you will use the knowledge built in Section 7.1 and Section 7.2 to analyze arguments and determine their truth value and validity. A logical argument is valid if its conclusion follows from its premises, regardless of whether those premises are true or false. You will then explore the truth tables for negation, conjunction, and disjunction and use these truth tables to analyze compound logical statements containing these connectives.
Interpret and Apply Negations, Conjunctions, and Disjunctions
The negation of a statement will have the opposite truth value of the original statement. When [latex]p[/latex] is true, [latex]\sim{p}[/latex] is false, and when [latex]p[/latex] is false, [latex]\sim{p}[/latex] is true.
Example 1
For each logical statement, determine the truth value of its negation.
a) [latex]p: 3+5=8[/latex]
[latex]p[/latex] is true because [latex]3+5[/latex] does equal 8; therefore, the negation of [latex]p[/latex], [latex]\sim{p}:3+5 \neq 8[/latex], is false.
b) [latex]q[/latex]: All horses are mustangs.
[latex]q[/latex] is false because there are other types of horses besides mustangs, such as Clydesdales or Arabians; therefore, the negation of [latex]q[/latex], [latex]\sim{q}[/latex] is true.
c) [latex]\sim{r}[/latex]: Baton Rouge is the capital of Louisiana.
[latex]\sim{r}[/latex] is false because Baton Rouge is the capital of Louisiana; therefore, the negation of [latex]\sim{r}[/latex], [latex]r[/latex], is true.
Exercise 1
For each logical statement, determine the truth value of its