# Logic

## Constructing Truth Tables


### Learning Objectives

After completing this section, you should be able to:
1. Interpret and apply negations, conjunctions, and disjunctions.
2. Construct a truth table using negations, conjunctions, and disjunctions.
3. 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  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 Statements and Quantifiers and Compound Statements 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  is true,  is false, and when  is false,  is true.
A conjunction is a logical and statement. For a conjunction to be true, both statements that make up the conjunction must be true. If at least one of the statements is false, the and statement is false.
A disjunction is a logical inclusive or statement, which means that a disjunction is true if one or both statements that form it are true. The only way a logical inclusive or statement is false is if both statements that form the disjunction are false.
In the next example, you will apply the dominance of connectives to find the truth values of compound statements containing negations, conjunctions, and disjunctions and use a table to record the results. When constructing a truth table to analyze an argument where you can determine the truth value of each component statement, the strategy is to create a table with two rows.  The first row contains the symbols representing the components that make up the compound statement. The second row contains the truth values of each component below its symbol. Include as many columns as necessary to represent the value of each compound statement. The last column includes the complete compound statement with its truth value below it.

### Construct Truth Tables to Analyze All Possible Outcomes

Recall from Statements and Questions that the negation of a statement will always have the opposite truth value of the original statement; if a statement  is false, then its negation  is true, and if a statement  is true, then its negation  is false. To create a truth table for the negation of statement , add a column with a heading of , and for each row, input the truth value with the opposite value of the value listed in the column for , as depicted in the table below.
Conjunctions and disjunctions are compound statements formed when two logical statements combine with the connectives “and” and “or” respectively.  How does that change the number of possible outcomes and thus determine the number of rows in our truth table? The multiplication principle, also known as the fundamental counting principle, states that the number of ways you can select an item from a group of  items and another item from a group with  items is equal to the product of  and . Because each proposition  and  has two possible outcomes, true or false, the multiplication principle states that there will be  possible outcomes: {TT, TF, FT, FF}.
The tree diagram and table in  demonstrate the four possible outcomes for two propositions  and .
A conjunction is a logical and statement. For a conjunction to be true, both the statements that make up the conjunction must be true. If at least one of the statements is false, the and statement is false.
A disjunction is a logical inclusive or statement. Which means that a disjunction is true if one or both statements that make it are true. The only way a logical inclusive or statement is false is if both statements that make up the disjunction are false.

###  Determine the Validity of a Truth Table for a Compound Statement

A logical statement is valid if it is always true regardless of the truth values of its component parts. To test the validity of a compound statement, construct a truth table to analyze all possible outcomes.  If the last column, representing the complete statement, contains only true values, the statement is valid.

### Check Your Understanding


### Key Terms

1. Truth table
2. Multiplication principle
3. Valid

### Key Concepts

1. Determine the true values of logical statements involving negations, conjunctions, and disjunctions.
        
          
      
2. Know how to construct a truth table involving negations, conjunctions, and disjunctions and apply the dominance of connectives to determine the truth value of a compound logical statement containing, negations, conjunctions, and disjunctions.
        
          
3. A logical statement is valid if it is always true. Know how to construct a truth table for a compound statement and use it to determine the validity of compound statements involving negations, conjunctions, and disjunctions.

### Video

1. Logic Part 2: Truth Values of Conjunctions: Is an "AND" statement true or false?
2. Logic Part 3: Truth Values of Disjunctions: Is an "OR" statement true or false?
3. Logic Part 4: Truth Values of Compound Statements with "and", "or", and "not"
4. Logic Part 5: What are truth tables? How do you set them up?
5. Logic Part 6: More on Truth Tables and Setting Up Rows and Column Headings