Contemporary Mathematics

# 2.3Constructing Truth Tables

Contemporary Mathematics2.3 Constructing Truth Tables

Figure 2.8 Just like solving a puzzle, a computer programmer must consider all potential solutions in order to account for each possible outcome. (credit: modification of work “Deadline Xmas 2010” by Allan Henderson/Flickr, CC BY 2.0)

### 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 $pp$ 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 $pp$ is true, $~p~p$ is false, and when $pp$ is false, $~p~p$ is true.

### Example 2.12

#### Finding the Truth Value of a Negation

For each logical statement, determine the truth value of its negation.

1. $pp$: $3+5=8.3+5=8.$
2. $qq$: All horses are mustangs.
3. $~r~r$: New Delhi is not the capital of India.

For each logical statement, determine the truth value of its negation.
1.
$\text~p$: $3 \times 5 = 14$.
2.
$\text~q$: Some houses are built with bricks.
3.
$r$: Abuja is the capital of Nigeria.

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.

### Example 2.13

#### Finding the Truth Value of a Conjunction

Given $p: 4+7=11,p: 4+7=11,$ $q: 11-3=7,q: 11-3=7,$ and $r: 7×11=77,r: 7×11=77,$ determine the truth value of each conjunction.

1. $p∧qp∧q$
2. $~q∧r~q∧r$
3. $~p∧q~p∧q$

Given $p$: Yellow is a primary color, $q$: Blue is a primary color, and $r$: Green is a primary color, determine the truth value of each conjunction.
1.
$p \wedge q$
2.
$q \wedge r$
3.
$\text{~}r \wedge p$

### Checkpoint

The only time a conjunction is true is if both statements that make up the conjunction are true.

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.

### Example 2.14

#### Finding the Truth Value of a Disjunction

Given $p: 4+7=11,p: 4+7=11,$ $q: 11-3=7,q: 11-3=7,$ and $r: 7×11=77,r: 7×11=77,$ determine the truth value of each disjunction.

1. $p∨qp∨q$
2. $~q∨r~q∨r$
3. $~p∨q~p∨q$

Given $p$: Yellow is a primary color, $q$: Blue is a primary color, and $r$: Green is a primary color, determine the truth value of each disjunction.
1.
$p \vee q$
2.
$\text{~}p \vee r$
3.
$q \vee r$

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.

### Example 2.15

#### Finding the Truth Value of Compound Statements

Given $p:4+7=11,p:4+7=11,$ $q:11-3=7,q:11-3=7,$ and $r:7×11=77,r:7×11=77,$ construct a truth table to determine the truth value of each compound statement

1. $~p∧q∨r~p∧q∨r$
2. $~p∨q∧r~p∨q∧r$
3. $~(p∧r)∨q~(p∧r)∨q$

Given $p$: Yellow is a primary color, $q$: Blue is a primary color, and $r$: Green is a primary color, determine the truth value of each compound statement, by constructing a truth table.
1.
$\text{~}q \wedge p \vee r$
2.
$p \vee q \wedge \text{~}r$
3.
$\text{~}(p \wedge r) \wedge q$

### 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 $pp$ is false, then its negation $~p~p$ is true, and if a statement $pp$ is true, then its negation $~p~p$ is false. To create a truth table for the negation of statement $pp$, add a column with a heading of $~p~p$, and for each row, input the truth value with the opposite value of the value listed in the column for $pp$, as depicted in the table below.

Negation
$pp$ $~p~p$
T F
F T

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 $nn$ items and another item from a group with $mm$ items is equal to the product of $mm$ and $nn$. Because each proposition $pp$ and $qq$ has two possible outcomes, true or false, the multiplication principle states that there will be $2×2=42×2=4$ possible outcomes: {TT, TF, FT, FF}.

The tree diagram and table in Figure 2.9 demonstrate the four possible outcomes for two propositions $pp$ and $qq$.

Figure 2.9

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.

Conjunction (AND) Disjunction (OR)
$pp$ $qq$ $p∧qp∧q$ $pp$ $qq$ $p∨qp∨q$
T T T T T T
T F F T F T
F T F F T T
F F F F F F

### Example 2.16

#### Constructing Truth Tables to Analyze Compound Statements

Construct a truth table to analyze all possible outcomes for each of the following arguments.

1. $p∨~qp∨~q$
2. $~(p∧q)~(p∧q)$
3. $(p∨~q)∧r(p∨~q)∧r$

Construct a truth table to analyze all possible outcomes for each of the following arguments.
1.
$p∧\text{~}q$
2.
$\text{~}(p∨q)$
3.
$(p∧\text{~}q)∨r$

### 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.

### Example 2.17

#### Determining the Validity of Compound Statements

Construct a truth table to determine the validity of each of the following statements.

1. $~p∧q~p∧q$
2. $~(p∧~p)~(p∧~p)$

Construct a truth table to determine the validity of each of the following statements.
1.
$p \vee \text{~}p$
2.
$\text{~}p \vee \text{~}q$

14.
A logical argument is _____ if its conclusion follows from its premises.
15.
A logical statement is valid if it is always _____.
16.
A _____ _____ is a tool used to analyze all the possible outcomes for a logical statement.
17.
The truth table for the conjunction, $p \wedge q$, has _____ rows of truth values.
18.

The truth table for the negation of a logical statement, $\text{~}p$, has _____ rows of truth values.

### Section 2.3 Exercises

For the following exercises, find the truth value of each statement.
1 .
$p$: $7 \times 3 = 21$. What is the truth value of $\text~p$ ?
2 .
$q$: The sun revolves around the Earth. What is the truth value of $\text~q$ ?
3 .
$\text~r$: The acceleration of gravity is $9.81\,$m/sec2. What is the truth value of $r$?
4 .
$s$: Dan Brown is not the author of the book, The Davinci Code. What is the truth value of $\text~{\text(\text{~}s)}$ ?
5 .
$t$: Broccoli is a vegetable. What is the truth value of $\text~{\text(\text{~}t)}$ ?
For the following exercises, given $p$: $1 + 2 = 3$, $q$: Five is an even number, and $r$: Seven is a prime number, find the truth value of each of the following statements.
6 .
$\text~q$
7 .
$p \wedge q$
8 .
$p \vee q$
9 .
$\text{~}p \vee \text{~}q$
10 .
$p \wedge \text{~}q$
11 .
$p \wedge r$
12 .
$q \wedge r$
13 .
$q \wedge \text{~}r$
14 .
$q \vee \text{~}r$
15 .
$\text{~}(p \wedge r)$
16 .
$p \vee q \wedge r$
17 .
$\text{~}p \vee (q \wedge r)$
18 .
$\text{~}(q \wedge r) \vee \text{~}p$
19 .
$q \vee r \vee p$
20 .
$\text{~}q \wedge r \wedge p$
For the following exercises, complete the truth table to determine the truth value of the proposition in the last column.
21 .
$p$ $q$ $r$ $\text{~}p$ $\text{~}p \vee q$ $(\text{~}p \vee q) \wedge r$
T T T
22 .
$p$ $q$ $r$ $\text{~}p$ $\text{~}p \wedge q$ $(\text{~}p \wedge q) \wedge r$
F T F
23 .
$p$ $q$ $r$ $\text{~}p$ $\text{~}r$ $\text{~}p \wedge q$$(\text{~}p \wedge q)\vee \text{~}r$
F F F
24 .
$p$ $q$ $r$ $\text{~}p$ $\text{~}r$ $\text{~}p \vee q$ $(\text{~}p \vee q) \vee \text{~}r$
F F F
For the following exercises, given $p{:}$ All triangles have three sides, $q{:}$ Some rectangles are not square, and $r{:}$ A pentagon has eight sides, determine the truth value of each compound statement by constructing a truth table.
25 .
$\text{~}r \wedge q \wedge p$
26 .
$\text{~}(q \wedge p) \vee r$
27 .
$\text{~}p \vee q \wedge r$
28 .
$\text{~}p \vee \text{~}q \vee r$

For the following exercises, construct a truth table to analyze all the possible outcomes for the following arguments.

29 .

$\text{~}q \wedge q$

30 .

$\text{~}p \vee \text{~}q$

31 .

$\text{~}p \wedge \text{~}q$

32 .

$p \wedge q \vee r$

For the following exercises, construct a truth table to determine the validity of each statement.

33 .

$\text{~}q \vee q$

34 .

$p \wedge \text{~}q$

35 .

$p \wedge q \vee \text{~}p$

36 .

$(p \wedge q) \vee (\text{~}p \wedge \text{~}q)$

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