2. LOGIC
Definition
the basis of all mathematical reasoning and all of
automated reasoning.
Rules of logic are used to distinguish between
valid and invalid mathematical arguments.
discrete mathematics
4. WHICH OF THESE
ARE
PREPOSITIONS?
Tokyo is the
capital of Japan
1 + 1 = 2
2 + 5 = 11
discrete mathematics
What time is it?
Read this carefully.
x + y = 12
5. • WE USE LETTERS TO DENOTE PROPOSITIONAL
VARIABLES (OR STATEMENT VARIABLES), THAT
IS, VARIABLES THAT REPRESENT
PROPOSITIONS, JUST AS LETTER ARE USED TO
DENOTE NUMERICAL VARIABLES.
• It uses the conventional letters such as p,q,r,s,...
• Truth values is denoted by T, if it's a true proposition, and
how do we represent
prepositions?
discrete mathematics
6. propositional
logic
• Area of logic that deals with prepositions
• First developed systematically by Greek
philosopher Aristotle more than 2300 years ago.
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august 2020
8. discrete mathematics
Let p be a proposition. The
negation of p, denoted by ¬p, is the
statement
"It is not the case that p"
The truth value of the negation of p,
¬p, is the opposite of the truth value
of p.
NEGATION
Definition 1
9. EXAMPLE 1
FIND THE NEGATION OF THE PROPOSITION
Michelle's PC
runs Linux.
SOLUTION
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It's not the case that
Michelle's Pc runs Linux.
Michelle's PC doesn't run
Linux.
PROBLEM
10. EXAMPLE 2
FIND THE NEGATION OF THE PROPOSITION
Vanessa's smartphone has at
least 32GB of memory.
SOLUTION
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It's not the case that Vanessa's
smartphone has at least 33GB of
memory.
PROBLEM
Vanessa's smartphone doesn't have
at least 32GB of memory.
11. august 2020
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Truth table for the negation of
proposition of p
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12. Let p and q be propositions. The
conjunction of p and q, denoted by
p ∧ q ,is the proposition.
"p and q"
The conjunction of p and q is true
when both p and q are true and
false if otherwise.
conjunction
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Definition 2
13. SAMPLE PROBLEM
FIND THE CONJUNCTION OF THE PROPOSITIONS P AND Q
WHERE P IS THE PROPOSITION:
Rachelle's PC has more than 50GB free hard disk space.
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AND Q IS THE PROPOSITION:
The processor in Rachelle's PC runs faster than 2 GHz.
14. SOLUTION
THE CONJUNCTION OF THESE PREPOSITIONS, P ∧ Q IS
THE PREPOSITION:
Rachelle's PC has more than 50GB free hard disk space AND the processor
in Rachelle's PC runs faster than 2 GHz.
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15. Truth table for the conjunction of two
propositions.
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Somethingtothinkabout
16. Let p and q be propositions. The
disjunction of p and q, denoted by
p ∨ q ,is the proposition.
"p or q"
The disjunction of p ∨ q is
false when both p and q are false
and is true if otherwise.
disjunction
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Definition 3
17. SAMPLE PROBLEM
FIND THE DISJUNCTION OF THE PROPOSITIONS P AND Q
WHERE P IS THE PROPOSITION:
Rachelle's PC has more than 50GB free hard disk space.
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AND Q IS THE PROPOSITION:
The processor in Rachelle's PC runs faster than 2 GHz.
18. SOLUTION
THE DISJUNCTION OF THESE PREPOSITIONS, P ∨ Q IS
THE PREPOSITION:
Rachelle's PC has more than 50GB free hard disk space OR the processor
in Rachelle's PC runs faster than 2 GHz.
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19. Truth table for the disjunction of two
propositions.
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Somethingtothinkabout
20. Let p and q be propositions. The
exclusive or of p and q, denoted by
p ⊕ q ,is the proposition that is true
when exactly one of p and q is
true and is false otherwise.
exclusive
or
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Definition 4
21. SAMPLE PROBLEM
FIND THE EXCLUSIVE OR OF THE PROPOSITIONS P AND Q
WHERE P IS THE PROPOSITION:
AND Q IS THE PROPOSITION:
Rachelle's PC has more than 50GB free hard disk space.
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The processor in Rachelle's PC runs faster than 2 GHz.
22. SOLUTION
THE EXCLUSIVE OR OR P AND Q,(P ⊕ Q) IS THE
PROPOSITION:
Rachelle's PC has at least 50GB free hard disk space , OR the processor
in Rachelle's PC runs faster than 2 GHz, but not both.
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23. Truth table for the Exclusive OR of
two propositions.
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Somethingtothinkabout
24. Let p and q be propositions. The
conditional statement p --> q is the
proposition, "if p, then q".
The conditional statement p --> q is
false when p is true and q is false,
and true otherwise.
CONDITIONAL
STATEMENT
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Definition 5
In the conditional statement p --> q, p is called the
hypothesis (or the antecedent or premise) and q is
called the conclusion (or the consequence)
25. "if p, then q"
p → q
august 2020 discrete mathematics
"if p, q"
"q if p"
"q when p"
"q follows from p"
"a necessary condition for p is q"
"a sufficient condition for q is p"
"p implies q"
"p only if q"
"q whenever p"
"q is necessary for p"
26. SAMPLE PROBLEM 1
LET P BE THE STATEMENT:
AND Q THE STATEMENT:
"Joy learns Discrete Mathematics."
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"Joy will find a good job."
EXPRESS THE STATEMENT P → Q INTO ENGLISH
27. SOLUTION
P → Q
"If Joy learns Discrete Mathematics, then she
will find a good job."
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28. SAMPLE PROBLEM 2
LET P BE THE STATEMENT:
AND Q THE STATEMENT:
"I am elected as the town Mayor."
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"I will lower the taxes and tuition fees."
EXPRESS THE STATEMENT P → Q INTO ENGLISH
29. SOLUTION 2
P → Q
"If I am elected as the town Mayor, then I will
lower the taxes and tuition fees."
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30. ANOTHER EXAMPLE
P → Q
"If you get 100 on the exam, then I will give you
1.0 grade."
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31. Truth table for the Conditional
Statement p → q.
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Somethingtothinkabout
33. Proposition:
Given p → q
converse
q → p
contrapositive
¬ q → ¬ p
inverse
¬ p → ¬q
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34. If you pass the exam, then you
will graduate on time.
converse
If you graduate on time, then you passed
the exam.
contrapositive
inverse
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Proposition:
p → q
If you do not graduate on time, then you
did not pass the exam.
If you don't pass the exam, then you don't
graduate on time.
36. • Let p and q be propositions. The biconditional statement
p ↔ q, is the proposition "p if and only if q."
• The biconditional statement p ↔ q is true when p and q
has the same truth vales, and false otherwise.
Definition 6
DISCRETE MATHEMATICS
• Biconditional statements are also called bi-implications.
BICONDITIONALS
37. WHAT ARE THE
WAYS TO EXPRESS
P ↔ Q?
"P is necessary and
sufficient for Q"
"If p and q, and
conversely"
"p iff q
38. SAMPLE PROBLEM 1
LET P BE THE STATEMENT:
AND Q THE STATEMENT:
"You can take the flight."
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"You buy a ticket."
EXPRESS THE STATEMENT P ↔ Q USING IFF
39. SOLUTION
THEN THE P ↔ Q STATEMENT:
"You can take the flight, if and only if you buy a
ticket."
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40. Truth table for the Biconditional
Statement p ↔ q.
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Somethingtothinkabout
Note :
p ↔ q has exactly the same
truth value as...
(p → q) ∧ (q → p)
