Download lec03_AT

Discrete Mathematical
‫الرياضيات المتقطعة‬
Example
OR
Q(x,y): x+y=x-y
a) Q(1,1): 2=0 False
b) Q(2,0): 2+0=2-0 True
c) Q(1,y): 1+y=1-y False(take any y<>0, x: y=1)
d) Q(x,2): x+2=x-2 False
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First: SOLUTION
Q1. (5 pts) Show that the following argument form is invalid:
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Q2. Use the truth table to show if the argument is valid.
" If this number is larger
than 2, then its square
is larger than 4."
" This number is not larger
than 2. "
The square of this
number is not larger
than 4.
p→q
p
q
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Predicates - multiple quantifiers
(Nested quantifiers)
To bind many variables, use many quantifiers!
Example: P(x,y) = “x > y”
 x P(x,y) c)
 xy P(x,y) b)
 xy P(x,y) a)
 x P(x,3) b)
a) True proposition
b) False proposition
c) Not a proposition
d) No clue
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Predicates - the meaning of multiple quantifiers
 xy P(x,y)
P(x,y) true for all x, y pairs.
 xy P(x,y)
P(x,y) true for at least one x, y pair.
 xy P(x,y)
For every value of x we can find a (possibly different)
y so that P(x,y) is true.
 xy P(x,y) There is at least one x for which P(x,y)
is always true.
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quantification order is not
commutative.
Predicates - the meaning of multiple quantifiers
N(x,y) = “x is sitting by y”
 xy N(x,y)
False
 xy N(x,y)
True
 xy N(x,y)
True?
 xy N(x,y)
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False
Multiple quantifiers (Examples)
1.  x  y, P(x,y):
For all x and for all y the relation P(x,y) is true.
If two numbers are integers then their product is
an integer.
2.  x  y, P(x,y):
For all x there is some y such that P(x,y) is true.
Every student has a favorite teacher
Note: here and below in all examples concerning
people, we shall assume that the domain is known
and will not represent it neither separately, nor
within the predicate expression.
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Multiple quantifiers (Examples)
3.  x  y, P(x,y):
There is some x such that for all individuals y the
relation P(x,y) is true.
Someone is loved by everybody
 x  y loves (y,x)
There is a professor that is liked by all students
4.  x  y, P(x,y):
There is some x and there is some y such that
P(x,y) is true.
Some students have favorite teachers
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Extra exmples for multiple
quantifiers
• xy P(x, y)
– “For all x, there exists a y such that P(x,y)”
– Example: xy (x+y == 0)
• xy P(x,y)
– There exists an x such that for all y P(x,y) is true”
– Example: xy (x*y == 0)
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Order of quantifiers
• xy and xy are not equivalent!
• xy P(x,y)
– P(x,y) = (x+y == 0) is false
• xy P(x,y)
– P(x,y) = (x+y == 0) is true
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