M ACM 101
Fundamentals of Logic
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M ACM 101
Fundamentals of Logic
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Back to Statements
Statements - cont’d…
A declarative sentence is an open statement if
1) it contains one or more variables, and
2) it is not a statement, but
3) it becomes a statement when the variables in it are
replaced by certain allowable choices.
E.g. q(x,y): The numbers y+2, x-y, and x+2y are even integers.
E.g.
We can quantify an open statement using:
existential quantifier: “For some x”: !x
universal quantifier: “For all x”: "x
q(5,2):
Therefore,
For some x, p(x) is
For some x,y, q(x,y) is
p(x): The number x+2 is an even integer.
p(5):
¬p(7):
p(6):
¬p(8):
M ACM 101
Fundamentals of Logic
¬q(4,7):
For some x, ¬p(x) is
For some x,y, ¬q(x,y) is
x in p(x):
x in !x,p(x):
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M ACM 101
Fundamentals of Logic
Example
Implicit Quantification
Consider the universe of all real numbers and the open
statements:
p(x): x ! 0
Sometimes statements (like those found in textbooks)
implicitly imply quantification:
q(x): x2 ! 0
r(x): x2-3x-4 = 0
s(x): x2-3 > 0
E.g. the trigonometric identity:
sin2x + cos2x = 1
really means:
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M ACM 101
Fundamentals of Logic
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M ACM 101
Fundamentals of Logic
Quantifier Examples
Examples
Consider the statements:
Consider the statements:
p(x): Student x likes Physics.
q(x): Student x is a genius.
r(x): Student x is happy.
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“If a quadrilateral is a rectangle then it has four equal angles.”
“If a quadrilateral has four equal angles, then it is a rectangle.”
Translate the following into quantified statements:
“Not every student likes Physics.”
“Every student who likes Physics is a genius, but not happy.”
“For every integer n we call n even if it is divisible by 2.”
“For every student who is a genius there is a student who likes Physics.”
M ACM 101
Fundamentals of Logic
Logical Equivalence &
Open Statements
As we might expect, the open statements p(x) and q(x) are
called logically equivalent, and we write
"x[p(x) # q(x)] when p(x) $ q(x) for all x.
Similarly, if p(x) % q(x) is true for all x then p(x)
logically implies q(x) and we write x[p(x) & q(x)].
Finally, for the statement "x[p(x) % q(x)]:
contrapositive:
converse:
inverse:
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M ACM 101
Fundamentals of Logic
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Example
Consider the universe of all integers and the open statements:
r(x): 2x+1 = 5
s(x): x2 = 9
M ACM 101
Fundamentals of Logic
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M ACM 101
Fundamentals of Logic
Logical Equivalences & Implications
Negating Statements
For a prescribed universe and any open statements p(x), q(x):
Rules for negating statements with one quantifier:
¬["x p(x)] #
!x[p(x) ' q(x)] &
[!x p(x) ' !x q(x)]
¬[!x p(x)] #
!x[p(x) ( q(x)] #
"x[p(x) ' q(x)] #
¬["x ¬p(x)] #
["x p(x) ( "x q(x)] &
M ACM 101
Fundamentals of Logic
Example
For the universe of all integers we have the following:
p(x): x is odd.
q(x): x2-1 is even.
Find the negation of the statement "x[p(x) % q(x)]
¬[!x ¬p(x)] #
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