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CS 2233 Discrete Mathematical Structures -- Carola Wenk
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CS 2233 -- Fall 2008
Discrete Mathematical Structures
1.1 Propositional Logic
Carola Wenk
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CS 2233 Discrete Mathematical Structures -- Carola Wenk
Propositions
Definition.
A^ proposition
is a sentence that is either
true (
T) or false (
F), but not both.
Examples:
Which of the following are propositions?
- The Alamo is located in San Antonio.• 4+2 = 42• UTSA is the best school in the world.• 23 = 6• 2
x^ = 6
- It is warm in San Antonio
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CS 2233 Discrete Mathematical Structures -- Carola Wenk
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Negation
Definition.
Let^ p
be a proposition. The
negation
(“not”) of
p, denoted by ¬
p, has the
opposite truth value than the truth value of p.Read ¬
p^ as: “not
p” or “It is not the case that
p”.
Truth Table:^ p^
¬p T^
F
F^
T
Examples:
Negate the following:
- “The Alamo is located in San Antonio.”» “The Alamo is not located in San Antonio”or “It is not the case that the Alamo is located in San Antonio”• Today is Monday» “Today is not Monday” or “It is not the case that today isMonday”
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CS 2233 Discrete Mathematical Structures -- Carola Wenk
Conjunction
Definition.
Let^ p
and^
q^ be propositions. The
conjunction
(“and”) of
p^ and
q, denoted by
p^ q, is true when both
p^ and
q^ are true and
is false otherwise.Read
p^ q
as: “
p^ and
q”.
Truth Table:^ p^
q^ p
q T^ T
T
T^ F
F
F^ T
F
F^ F
F
Examples:
Find the conjunction of
p^ and
q:
-^ p: “It is sunny today.”
q: “Today is Monday.”
» “It is sunny today and today is Monday.”The conjunction is true on sunny Mondays (
TT) but it is false
on any non-sunny day (
FT^ or
FF) and it is false on any other
day but Monday (
TF^ or
FF).
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CS 2233 Discrete Mathematical Structures -- Carola Wenk
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Disjunction
Definition.
Let^ p
and^
q^ be propositions. The
disjunction
(“inclusive or”) of
p^ and
q,
denoted by
p^ q
, is false when both
p^ and
q
are false and is true otherwise.Read
p^ q
as: “
p^ or^
q”.
Truth Table:^ p^
q^ p
q T^ T
T
T^ F
T
F^ T
T
F^ F
F
Examples:
Find the disjunction of
p^ and
q:
-^ p: “It is sunny today.”
q: “Today is Monday.”
» “It is sunny today or today is Monday.”The disjunction is true on sunny Mondays (
TT) and on
Mondays (
FT^ or
TT) and on sunny days (
TF^ or
TT). It is only
false on non-sunny days that are not Mondays (
FF).
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CS 2233 Discrete Mathematical Structures -- Carola Wenk
Exclusive Or
Definition.
Let^ p
and^
q^ be propositions. The
exclusive or
(“xor”) of
p^ and
q, denoted by
p⊕q, is true when exactly one of
p^ and
q^ is
true, and false otherwise.Read
p⊕q
as: “
p^ xor
q”.
Truth Table:^ p^
q^
p⊕q T^ T
F
T^ F
T
F^ T
T
F^ F
F
Where is the difference between or and xor? • “Students who have taken calculus or biology can take thisclass.” Is this
p^ q
or^ p⊕
q^?
- The use of “or” in English is usually inclusive (i.e.,
- How can we make this statement exclusive (i.e.,
» “Students who have taken calculus or biology, but not both,can enroll in this class.”Note that “either….or” is supposed to be exclusive, but we oftendon’t use it in the correct way in English.
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CS 2233 Discrete Mathematical Structures -- Carola Wenk
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Conditional Statement
Definition.
Let^ p
and^
q^ be propositions. The
conditional statement p
→q^ is false when
p^ is
true and
q^ is false, and true otherwise.
p^ is
called the
hypothesis
and^
q^ the
conclusion
Truth Table:^ p^
q^
p→q T^ T
T
T^ F
F
F^ T
T
F^ F
T
Read
p→q
as: “if
p,^ then
q”^
“p^ implies
q”
“p^ only if
q” ….. many more examples in the book.
Examples: • “If I am elected, then I will lower taxes.”»^ p→
q^ with
p^ “elected” and
q^ “taxes”
-^ p: “It rains.”
q: “We get wet.” » “If it rains, then we will get wet.”» “We will get wet whenever it rains.”» “It rains only if we get wet.”
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CS 2233 Discrete Mathematical Structures -- Carola Wenk
Biconditional Statement
Definition.
Let^ p
and^
q^ be propositions. The
biconditional statement
(“iff”)
p→q
is true
when
p^ and
q^ have the same truth value, and false otherwise.
Truth Table:^ p^
q^
p↔q T^ T
T
T^ F
F
F^ T
F
F^ F
T
Read
p↔q
as: “
p^ if and only if
q”
“p iff q
Example: • “You can take the flight if and only if you buy a ticket.”
