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Math 101 – Final Exam “Walk-through” Review of Logic
Logic Symbol Summary
Operator
NOT
AND
Symbols
~p
pq
Description Negation Conjunction
Truth Table
p
T
F
~p
F
T
p
T
T
F
F
q
T
F
T
F
OR
pq
Disjunction
pq
T
F
F
F
p
T
T
F
F
q
T
F
T
F
IMPLIES
pq
Conditional
pq
T
T
T
F
p
T
T
F
F
q
T
F
T
F
p→q
T
F
T
T
Venn
Diagram
Equivalent Statements: Their truth tables have the same _______________________
 means “is equivalent to”
DeMorgan’s Law:
a) ~ (p  q)  ~ p  ~ q (negation of AND)
b) ~ (p  q)  ~ p  ~ q (negation of OR)
Ex a Use truth tables to demonstrate ~ (p  q) is equivalent to ~ p  ~ q
p
q
Negations of the other 2 operators
a) ~ (~p) 
(negation of NOT)
b) ~ (pq) 
(negation of IMPLIES)
Ex b Use truth tables to test whether ~ (pq) is equivalent to p  ~q
p
q
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Conditionals: “If” vs. “Only If”
If
p
then
q
(Intuitive order)
The word “If” (alone) points to the _______________________ immediately after it.
Ex:
You are not a vegetarian if you eat meat.
Only if
q
then
p

p
only if
q
(Less intuitive order)
The phrase “Only if” points to the ______________________immediately after it.
Ex: You can graduate only if you pass math.
Ex
Give the premise and conclusion. Then arrange statements into “if p, then q” format.
1) You are not a vegetarian if you eat meat.
2) I ride my bike only if it’s not raining.
3) Only if you are committed will you stay married.
Biconditional: “If and Only If”
Symbolically: (pq)  (qp)
Shortcut: p  q
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Variations on Conditionals and Equivalance
Conditional Statement: pq

Contrapositive: ~ q~ p
Inverse: ~ p  ~q

Converse: qp
Ex
“I ride my bike only if it’s not raining. ”
1) Write the contrapositive, inverse & converse
2) Write the negation of the statement (hint: see page 1, negations of other operators)
Necessary and Sufficient Conditions
Cheap Trick:
 Sufficient:
Ex
Translate to symbols: Being a chimp is sufficient for not being a monkey.
Venn Diagram:

Necessary:
Ex
Translate to symbols: Not being a monkey is necessary for being an ape
Venn Diagram:
Ex
For each of the statements 1) Assign “p” and “q” to phrases, 2) Translate to symbols,
3) Rewrite the statement in “If p, then q” format, and 4) show the Venn Diagram
a) Warm weather is sufficient for no snow.
b) Clouds are necessary for snow.
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Valid Arguments
Some, All, and None Using Venn Diagrams
Ex
Determine the validity using a Venn Diagram.
1. No vegetarian owns a gun.
2. All policemen own guns
Therefore, no policeman is a vegetarian
Bonus: Based on your Venn Diagram, write an “If p then q” conditional statement
Bonus2: Based on your Venn Diagram, write statements using “necessary” & “sufficient”
Ex
Determine the validity using a Venn Diagram
1. All poets are eloquent.
2. Some poets are wine connoisseurs
Therefore some wine connoisseurs are eloquent
Determining Validity with Truth Tables
Hypothesis 1: If I’m not old enough, I can’t drink
Hypothesis 2: I’m old enough
Conclusion: Therefore, I can drink
Notation:
Use a truth table to determine whether the argument: above is valid.
p
q
1.
T
T
2.
T
F
3.
F
T
4.
F
F