CS103
HO#7
SlidesRules of Inference
April 5, 2010
1
CS103
Mathematical Foundations of Computing
4/5/10
Homework can be submitted in the filing cabinet in the lobby
near my office. There is a drawer marked CS103.
Notice that implication in a sentence like
x (P(x)
Q(x))
seems quite natural.
But what does
x (P(x)
Q(x))
mean?
It's better to write the second sentence as
x (¬P(x)
Q(x))
Translations
S(x): x is a student in this class
C(x):
x has visited Canada
M(x):
x has visited Mexico
Some student in this class has visited Mexico.
x ( S(x)
M(x) )
or is it
x ( S(x)
M(x) )
Every student in the class has visited Canada or Mexico.
x (S(x)
[ C(x)
M(x) ] )
Multiple Quantifiers
x
y (Boy(x)
Girl(y)
Likes(x, y))
x
y [(Tet(x)
Dodec(y))
Smaller(x, y)]
x
y ((Cube(x)
Cube(y))
. . .)
Could x and y be the same cube?
Multiple Quantifiers
x
y (Boy(x)
Girl(y)
Likes(x, y))
x
y [(Tet(x)
Dodec(y))
Smaller(x, y)]
x
y ((Cube(x)
Cube(y)
x ≠ y)
. . .)
x
y (Cube(x)
Cube(y)
x ≠ y)
Universe of Discourse
HasTaken(a, b)
a
has taken class
b
If we write
x
y HasTaken (x, y)
, it is understood
that the quantifiers operate over the appropriate
domains.
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CS103
HO#7
SlidesRules of Inference
April 5, 2010
2
x
y HasTaken (x, y)
x
y HasTaken (x, y)
x
y HasTaken (x, y)
x
y HasTaken (x, y)
y
x HasTaken (x, y)
y
x HasTaken (x, y)
Q(x,y) :
x + y = 0
y
x Q(x, y)
x
y Q(x, y)
Summary of Quantifiers in ArityTwo Predicates
Statement
When true?
When false?
x
y P(x,y)
P(x,y) is true for all pairs (x,y)
There is a pair (x,y) for which
P(x,y) is false
x
y P(x,y)
For every x, there is a y for which
There is an x such that P(x,y)
P(x,y) is true
is false for every y
x
y P(x,y)
There is an x for which P(x,y) is
For every x, there is a y for
true for every y
which P(x,y) is false
x
y P(x,y)
There is a pair (x,y) for which
P(x,y) is false for all pairs (x,y)
P(x,y) is true
y
x P(x,y)
y
x P(x,y)
What if we wanted to test alternatives to determine
if
x
y P(x, y)
is true?
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 Fall '09
 Logic, Modus ponens, kitchen table, Justification Justification Justification

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