The Logic of Quantified Statements Section 23 Multiple Quantifiers Multiple Quantifiers A statement may contain multiple quantifiers or When multiple quantifiers are encountered treat them as they come in order ID: 415276
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Slide1
Chapter 2
The Logic of Quantified StatementsSlide2
Section 2.3
Multiple Quantifiers Slide3
Multiple Quantifiers
A statement may contain multiple quantifiers, ∀,∃ or ∃, ∀.
When multiple quantifiers are encountered treat them as they come, “in order”
Example
∀
x
∈D, ∃
y
∈E such that
x
and
y
satisfy
P(x,y
)
Find
x
from D where “a”
y
from E satisfies property
P(x,y
)
∃
x
∈D, ∀
y
∈E
x
and
y
satisfy
P(x,y
)
Given
x
in D, any
y
from E must satisfy
P(x,y
) Slide4
∀∃ Statement
Show that the following is true: “For all triangles in
x
, there is a square
y
such that
x
and y have the same color.”Slide5
∃∀ Statement
Show that the following is true: “There is a triangle
x
such that for all circles
y
,
x
is to the right of y.”Slide6
Interpreting Multiply-Quantified Statements
Write informal statement and give its truth:
∃item I such that ∀Students S, S chose I
There is an item that all students selected. (true: Pie)
∀students S and ∀stations Z, ∃ item I in Z such that S chose I.
Every student chose at least one item from every station. (false: Yuen didn’t get a salad).Slide7
Informal to Formal Examples
A reciprocal of a real number a is a real number
b
such that
ab
= 1.
Translate the following
Every nonzero real number has a reciprocal.∀nonzero real number
u, ∃a real number v such that u
*
v
= 1.
There is a real number with no reciprocal.
∃a real number
c
such that ∀real numbers
d
,
c
*
d
≠ 1Slide8
Example
Translate the following to formal:
“There is a smallest positive number.”
∃a positive integer
m
such that ∀ positive integers
n
, m ≤
n.Is this true? Is there a positive integer such that it is equal or less than any other positive integer.Yes…. 1 works.Slide9
Negations of Multiply-Quantified Statements
Multiply-Quantified statements may be negated as is the case with simple quantified statements from 2.2.
~(∀
x
in D,
Q(x
) ) ≡∃
x ∈D such that ~
Q(x)~(∃x ∈D such that
Q(x
)) ≡ ∀
x
∈D, ~
Q(x
)
~(∀
x
in D, ∃
y
in E such that
P(x,y
))
First: ∃
x
in D such that ~(∃
y
in E such that
P(x,y
))
Final: ∃
x
in D such that ∀
y
in E, ~
P(x,y
)
~(∃
x
in D, ∀
y
in E such that
P(x,y
))
First: ∀
x
in D such that ~(∀
y
in E such that
P(x,y
))
Final: ∀
x
in D such that ∃
y
in E, such that ~
P(x,y
)Slide10
Example
Write a negation for the following statements:
For all squares
x
, there is a circle
y
such that
x and
y have the same color.First: ∃a square x such that ~(∃a circle
y
such that
x
and
y
have the same color)
Final: ∃a square
x
such that ∀circles
y
,
x
and
y
do not have the same color.
Negation is true. sq e is black and no circle is black.Slide11
Example
Write a negation for the following statements:
There is a triangle
x
such that for all squares
y
,
x is to the right of
y.First: ∀triangles x, ~(∀squares
y
,
x
is to the right of
y
)
Final: ∀triangles
x
, ∃ a square
y
such that
x
is not to the right of
y
.