Section 3'2: Truth Tables for Negation, Conjunction, and Disjunction

1
Section 3.2 Truth Tables for Negation,
Conjunction, and Disjunction

• Dr. Fred Butler
• Math 121 Fall 2004

2
Truth Tables

• A truth table is used to determine when a
  compound statement is true or false.
• They are used to break a complicated compound
  statement into simple, easier to understand
  parts.

3
Truth Table for Negation

• If P is a true statement then its negation P
  or not P is false. If P is false, then P
  is true.
• This is summarized in the truth table for
  negation below.

4
Four Possible Cases

• When a compound statement involves two simple
  statements P and Q, there are four possible cases
  for the combined truth values of P and Q.

5
When is a Conjunction True?

• Suppose I tell the class, You can retake the
  last exam and you can turn in this lab late.
• Let P be You can retake the last exam and Q
  You can turn in this lab late.
• Which truth values for P and Q make it so I kept
  my promise, to the class?

6
When is a Conjunction True? contd.

• P You can retake the last exam.
• Q You can turn this lab in late.
• There are four possibilities.
• 1. P true and Q true, then is
  true.
• 2. P true and Q false, then is
  false.
• 3. P false and Q true, then is
  false.
• 4. P false and Q false, then is
  false.

7
Truth Table for Conjunction

• This information is summarized in the truth table
  for conjunction below.

8
PRS Question 2.3

• What is the truth value of the statement, WVU is
  in Arizona and Morgantown is in West Virginia?
• 1. True 2. False

9
When is Disjunction True?

• Suppose I tell the class that for this unit you
  will receive full credit if You do the homework
  quiz or you do the lab.
• Let P be the statement You do the homework
  quiz, and let Q be the statement You do the
  lab.
• Which truth values for P and Q would I consider
  that you did what I said, which is
  to receive full credit for this unit?

10
When is Disjunction True? contd.

• P You do the homework quiz.
• Q You do the lab.
• There are four possibilities
• 1. P true and Q true, then is
  true.
• 2. P true and Q false, then is
  true.
• 3. P false and Q true, then is
  true.
• 4. P false and Q false, then is
  false.

11
Truth Table for Disjunction
12
PRS Question 2.4

• What is the truth value of the statement, WVU is
  in Arizona or Morgantown is in West Virginia?
• 1. True 2. False

13
Constructing a Truth Table

• Study the compound statement and determine its
  type (negation, disjunction, etc.), to determine
  where the final truth table will appear.
• Complete the columns under the simple statements
  (P, Q, R) and their negations (P, Q, R) within
  parentheses. If there are nested parentheses,
  work from the innermost outward.
• Complete the column under the connective within
  the parentheses.
• Complete the column under any remaining
  statements and their negations.
• Complete the column under any remaining
  connectives.

14
An Example of Constructing a Truth Table

• In the following slides, we will apply this
  method to construct a truth table for the
  statement

15
An Example contd.
16
An Example contd.
17
An Example contd.
18
An Example contd.
19
An Example contd.
20
An Example contd.
21
An Example contd.
22
Constructing Truth Tables with Three Simple
Statements

• So far all the compound statements we have
  considered have contained only two simple
  statements (P and Q), with only four true-false
  possibilities.

23
Constructing Truth Tables with Three Simple
Statements contd.

• When a compound statement consists of three
  simple statements (P, Q, and R), there are now
  eight possible true-false combinations.

24
Constructing Truth Tables with Three Simple
Statements contd.
25
A Three Statement Example

• Let us construct a truth table for the statement
  using the same techniques as
  before.

26
A Three Statement Example contd.
27
A Three Statement Example contd.
28
A Three Statement Example contd.
29
A Three Statement Example contd.
30
A Three Statement Example contd.
31
A Three Statement Example contd.
32
Lecture Summary

• A truth table is used to determine when a
  compound statement is true or false.
• A conjunction is true only when both P
  and Q are true.
• A disjunction is true when P is true, Q
  is true, or both P and Q are true.
• When a compound statement consists of 2 simple
  statements, there are 4 possible true-false
  combinations with 3 simple statements, there are
  8 true-false combinations.

33
Homework

• Do problems from Section 3.2 of the textbook.
• Print lecture notes.