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College Math
SECTION 3.2: TRUTH TABLES FOR NEGATION, CONJUNCTION, AND DISJUNCTION
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.
Truth Table for Negation
As you can see “P” is a true statement then its negation “~P” or “not P” is false.
If “P” is false, then “~P” is true.
Case 1
Case 2
P
T
TF
F
~P
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.
P Q
Case 1
Case 2
Case 3
Case 4
T
TT
T
F
F
F
F
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 be “You can turn in this lab late.”
Which truth values for P and Q make it so that I kept my promise, P Λ Q to the class?
When is a Conjunction True? cont’d.
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 P Λ Q is true.
2. P true and Q false, then P Λ Q is false.
3. P false and Q true, then P Λ Q is false.
4. P false and Q false, then P Λ Q is false.
Truth Table for Conjunction
P QCase 1
Case 2
Case 3
Case 4
T
T
F
F
T
F
T
F
T
F
F
F
P Λ Q
3.2 Question 1 W
hat is the truth value of the statement, “Caracas is in Venezuela AND Bogota is in Italy”?
1. True 2. False
When is Disjunction True?S
uppose 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.”
In this case a “truth” is equal to receiving full credit
When is Disjunction True? cont’d.
P: “You do the homework quiz.”
Q: “You do the lab.”
There are four possibilities:
1. P true and Q true, then P V Q is true.
2. P true and Q false, then P V Q is true.
3. P false and Q true, then P V Q is true.
4. P false and Q false, then P V Q is false.
Truth Table for Disjunction
P QCase 1
Case 2
Case 3
Case 4
T
T
F
F
T
F
T
F
T
T
T
F
P V Q
3.2 Question 2W
hat is the truth value of the statement, “Caracas is in Venezuela or Bogota is in Italy”?
1. True 2. False
Truth Table SummaryYou can remember the truth tables for ~ (not),
Λ (and), and, V(or) by remembering the following:
~(not) - Truth value is always the opposite
Λ(and)-Always false, except when both are true
V(or) - Always true, except when both are false
Making a Truth Table Example
Let’s look at making truth tables for a statement
involving only ONE Λ or V of simple statements P and Q and possibly negated simple statements ~P and ~Q.
For example, let’s make a truth table for the statement ~P V Q
Truth Table for ~P V Q
T
T
F
F
T
F
T
F
P ~PQ Q
Opposite of Column 1
F
F
T
T
Same as Column 2
T
F
T
F
T
F
T
T
FinalAnswercolumn
V
Another Example: P Λ ~Q
T
T
F
F
T
F
T
F
P PQ ~Q
Same as Column 1
T
T
F
F
Opposite of Column 2
F
T
F
T
F
T
F
F
FinalAnswercolumn
Λ
3.2 Question 3W
hat is the answer column in the truth table of the statement ~P Λ ~Q ?
1. T 2. T 3. F
F F F
F T F
F F T
~P Λ ~Q Stop Day 1
T
T
F
F
T
F
T
F
P ~PQ ~Q
Opposite of Column 1
F
F
T
T
Opposite of Column 2
F
T
F
T
F
F
F
T
FinalAnswercolumn
Λ
More Complicated Truth TablesN
ow suppose we want to make a truth table for a more complicated statement,
(P V~Q) V (~PΛQ)
We set the truth table up as before.
Our final answer will go under the most dominant connective not in parentheses
(the one in the middle)
More Complicated Truth Tables
Final Answer
T
T
F
F
Opposite of
Column 1
Opposite of
Column 2
Same as Column 2
Same as Column 1
F
T
F
T
OR
T
T
F
T
F
F
T
T
T
F
T
F
AND
F
F
T
F
T
T
T
T
More Complicated Truth TablesN
ow let’s make a truth table for
(P V ~Q) Λ (~P Λ Q)
Each of the statements in parentheses
( P V ~Q) and (~P Λ Q) are just like the statements we did previously, so we fill in their truth tables as we just did.
P Q (P ~Q) (~P Q)T T
T F
F T
F F
More Complicated Truth Tables
Final Answer
T
T
F
F
Opposite of
Column 1
Opposite of
Column 2
Same as Column 2
Same as Column 1
F
T
F
T
OR
T
T
F
T
F
F
T
T
T
F
T
F
AND
F
F
T
F
F
F
F
F
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.
P Q
Case 1 T T
Case 2 T F
Case 3 F T
Case 4 F F
Constructing Truth Tables with Three Simple Statements cont’d.
When a compound statement consists of three simple statements (P, Q, and R), there are now eight possible true-false combinations.
Constructing Truth Tables with Three Simple Statements cont’d.
P Q R
Case 1 T T T
Case 2 T T F
Case 3 T F T
Case 4 T F F
Case 5 F T T
Case 6 F T F
Case 7 F F T
Case 8 F F F
A Three Statement Example
Lets construct a truth table for the statement (P V Q) Λ ~R using the same techniques as before.
Remember, there are not more possible combinations because we added a third statement
A Three Statement ExampleP Q R (P Q) ~R
T T T
T T F
T F T
T F F
F T T
F T F
F F T
F F F
Final Answer
T
T
T
T
F
F
F
F
T
T
F
F
T
T
F
F
F
T
F
T
F
T
F
T
T
T
T
T
T
T
F
F
F
T
F
T
F
T
F
F
Practice•D
etermine the Truth Value for the statement IF:• P is true, Q is false, and R is true
(~ P V ~ Q) Λ (~R V ~ P)
Practice•T
ranslate into symbols. Then construct a truth table and indicate under what conditions the compound statement is TRUE.
•Tanisha owns a convertible and Joan does not own a Volvo.
Practice•C
onstruct a Truth Table for the following compound statement: R V(P Λ ~ Q)
DeMorgans Law (this guy again?)
More Complicated Truths; Quantifiers
•Quantifiers- Give an Amount to a statement
•Examples;
• All
• No/None
• Some
• Half
• At least one
•This makes a Negation (~) more difficult to define• Find the Negation of;
• Some Do
• All do
• None do
• At least one
Negations of Quantifiers•S
ome do•A
ll do •N
one do•A
t least one does
•None do (All do not)
•Some do Not (Not all do)
•Some do (None do not)
•None do
Examples of Negations with quantifiers1. Some girls play soccer2. All boys are immature
3. No students read books
4. At least one person likes anchovies
•No Girls play soccer
•Not all boys are immature (some are not immature)
•Some students read books
•No one likes anchovies
