section 3.2: truth tables for negation, conjunction, and disjunction
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Section 3.2: Truth Tables for Negation, Conjunction, and Disjunction. Math 121. 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. - PowerPoint PPT PresentationTRANSCRIPT
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Section 3.2: Truth Tables for Negation, Conjunction, and
Disjunction
Math 121
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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.
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Case 1
Case 2
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.
P
T
TF
F
~P
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Four Possible CasesWhen 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
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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?
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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.
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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
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3.2 Question 1
What is the truth value of the statement, “U of M is in Ann Arbor and Ann Arbor is in West Virginia”?
1. True 2. False
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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.”
For which truth values of P and Q would I say that you did what I said, which is PVQ to receive full credit for this unit?
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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.
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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
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3.2 Question 2
What is the truth value of the statement, “WVU is in Arizona or Morgantown is in West Virginia”?
1. True 2. False
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Truth Table Summary
You 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
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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 ~PVQ
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Truth Table for ~PVQ
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
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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
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3.2 Question 3
What 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
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~P Λ ~Q
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
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More Complicated Truth Tables
Now suppose we want to make a truth table for a more complicated statement,
(PV~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)
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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
T
T
T
T
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More Complicated Truth Tables
Now 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.
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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
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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
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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.
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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
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A Three Statement Example
Let,s construct a truth table for the statement (P V Q) Λ ~R using the same techniques as before.
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P 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
A Three Statement Example
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
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Practice
• Determine the Truth Value for the statement IF:
• P is true, Q is false, and R is true
(~ P V ~ Q) Λ (~R V ~ P)
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Practice
• Translate into symbols. Then construct a truth table and indicate under what conditions the compound statement is TRUE.
• Nathan owns a convertible and Joe does not own a Volvo.
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Practice
• Construct a Truth Table for the following compound statement: R V(P Λ ~ Q)