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Math 170 Project
Section 3.1 #25,#26,#27
Section 3.3 #12
By: Marlene Reyna, Mayra Aguilar,
Guadalupe Esquivel
#25
Rewrite each of the following statements in the two
forms “ _____ x, _____” and “ x, if _____, then
_____” or in the two forms “ x and y, _____, _____”
and “ x and y, if _____, then ______.”
Symbols used:
• Universal quantifier reads as “for all”
Ex. Another way to express “All human beings are mortal”, is to write
(“
human beings x, x is mortal.
1.Domain of x
___________
x, _________”) 2. Rewrite the statement but
Or
replace “all human beings” with
“x” and treat x as if it was
singular.
x, if x is a human being, then x is mortal.
(If and then statement)
(“ x, if _______________, then _________”)
Rewrite each of the following statements in the two forms “
_____ x, _____”
and “ x, if _____, then _____” or in the two forms “ x and y, _____, _____”
and “
x and y, if _____, then ______.”
a. The reciprocal of any nonzero fraction is a fraction.
•
nonzero fractions x, the reciprocal of x is a fraction. 2. Rewrite original statement
1. What is the
domain of x?
•
but replace “any nonzero
fraction” with “x”.
x, if x is a nonzero fraction, then the reciprocal of x is a fraction.
1. Say what x is based on
the domain
b. The derivative of any polynomial function is a polynomial function.
•
polynomial functions x, the derivative of x is a polynomial function.
2. Rewrite original
statement but replace
“any polynomial
function” with “x”.
1. What is the
domain of x?
• x, if x is a polynomial function, then the derivative of x is a polynomial function.
1.Say what x is based
on the domain.
c. The sum of the angles of any triangle is 180°.
•
triangles x, the sum of the angles of x is 180°.
1.What is the domain of x?
•
2. Rewrite original
statement but replace
“any triangle” with “x”.
x, if x is a triangle , then the sum of the angles of x is 180°.
1. Say what x is based on the domain.
d. The negative of any irrational number is irrational.
•
irrational numbers x, the negative of x is irrational.
1.What is the
domain of x?
•
2. Rewrite the original
statement but replace
“any rational number”
with “x”.
x, if x is an irrational number, then the negative of x is irrational.
1.Say what x is based on
the domain.
e. The sum of any two even integers is even.
•
even integers x and y, the sum of x and y is even.
1.What is the domain of x and y?
•
2. Rewrite the
original statement
but replace “any two
even integers” with
“x and y”
x and y, if x and y are even integers, then the sum of x and y is even.
1.Say what x and y are based on their domain.
f. The product of any two fractions is a fraction.
•
fractions x and y, the product of x and y is a fraction.
1.What is the domain of x and y?
•
2. Rewrite the original
statement but replace
“any two fractions”
with “x and y.”
x, if x and y are fractions, then the product of x and y is a fraction.
1. Say what x and y are
based on their domain.
Charles Sanders Peirce
• Born on September 10,1839 in
Cambridge, MA.
• American Philosopher, logician, and
engineer
• Known as “the father of pragmatism”
• Wrote on a wide range of topics
which included mathematical logic,
physics, astronomy, psychology, etc.
• He introduced the formal concept of
quantifier into symbolic logic in the
late nineteenth century.
• Died from cancer on April 19, 1914 in
Pennsylvania
Section 3.1
#27
Refer to the picture of Tarski’s world given in Example
3.1.13. Let Above(x, y) mean that x is above y (but
possibly in a different column). Determine the truth or
falsity of each of the following statements. Give
reasons for your
answers.
a. ∀u, Circle(u) →Gray(u).
b. ∀u, Gray(u) →Circle(u).
c. ∃y such that Square(y) ∧ Above(y,d).
d. ∃z such that Triangle(z) ∧ Above(f,z).
#26
Consider the statement, “All integers are
rational but some rational numbers are not
integers.”
• A. Write this statement in the form.
• “∀ x, if___ then ____, but _____ ∃x such that ____.”
• “∀ x, if it’s an integer then it is rational, but some
rational numbers ∃x such that are not all integers.”
Let Ratl(x) be “x”is a rational number” and
Intl(x) be “x is an integer.”
• Write the given statement formally using only the
symbols Ratl(x), Intl(x), V, ∃, ^, ~, ->.
• B. ∀x [Intl(x0  Ratl(x) ] ^∃x [Ratl(x) ^
~Intl(x)]
Tarski’s World
a
b
c
e
g
d
f
h
i
j
k
The program for Tarski’s World
provides pictures of blocks of
various sizes, shapes, and
colors, which are located on a
grid. To the left is a picture of
an arrangement of objects in a
two-dimensional Tarski world.
The configuration can be
described using
logical operators and—for the
two-dimensional version—
notation
a. ∀u, Circle(u) →Gray(u).
All circles are gray
a
False
b
c
e
g
d
A counter example
is b, which is a black
circle.
f
h
i
j
k
b. ∀u, Gray(u) →Circle(u).
All gray shapes are circles
a
True
b
c
e
g
d
All the gray
shapes are
circles
f
h
i
j
k
c. ∃y such that Square(y) ∧ Above(y,d).
There exist a y that is a square and that square
is above d (but possibly in a different column)
a
b
c
e
g
False
d
Look at the grid and
find d. There are no
squares above d’s
row
f
h
i
j
k
d. ∃z such that Triangle(z) ∧ Above(f,z).
There exist a z, which is a Triangle and f is above
that triangle(but possibly in a different column)
a
b
c
e
g
True
d
f
h
i
j
k
Look at the
grid and find f.
Then look for
the triangle
below it. You
will see that f
is above
triangle g.
Section 3.3
#12
Let D = E = {−2,−1,0,1,2}. Write
negations for each of the following
statements and determine which is true,
the given statement or its negation.
a. ∀x in D,∃y in E such that x + y = 1.
b. ∃x in D such that ∀y in E, x + y = −y.
c. ∀x in D, ∃y in E such that x y ≥ y.
d. ∃x in D such that ∀y in E, x ≤ y.
Negation
Symbol: ~
The negation of a statement can be found by using De Morgan’s
Law
~(p ∧ q) = ~p v ~q
or
~(p v q) = ~p ∧ ~q
“The negation of p and q is not p or not q”
The negation of a quantifier can be found using the same Law.
~(∀x)= ∃x
and
~(∃x)= ∀x
And everything that follows will be negated the same way a
statement is negated
Let D = E = {−2,−1,0,1,2}.
a. ∀x in D, ∃y in E such that x + y = 1.
Determine which is true. Original or Negation?
~(∀x in D,∃y in E such that x + y = 1)
Negation: ∃x in D, such that ∀y in E x + y ≠ 1.
“There exists an x in D , such that all y’s in E make x + y ≠ 1”
Let x=-2
When you plug x=-2 into the original statement you get:
-2 + y = 1
y= 3
3 is not in E
Hence, the original statement is False.
That negation is true
Let D = E = {−2,−1,0,1,2}.
b. ∃x in D such that ∀y in E, x + y = −y.
Determine which is true. Original or Negation?
~(∃x in D such that ∀y in E, x + y = −y)
Negation: ∀ x in D, ∃y in E such that x + y ≠ −y
“For all x in D, there exist a y in E such that x + y ≠ −y
Let x= 1 and y= 1
When you plug x and y into the original statement you get:
1+ 1 = -1
2 ≠ -1
Hence, the original statement is false.
The negation is true.
Let D = E = {−2,−1,0,1,2}.
c. ∀x in D, ∃y in E such that x y ≥ y.
Determine which is true. Original or Negation?
~(∀x in D, ∃y in E such that x y ≥ y)
Negation: ∃x in D, such that ∀y in E x y < y
“There exists an x in D such that all y’s in E makes x y < y”
Let x=1 and y= -2
When you plug x and y into the negation you get:
(1)(-2)< -2
-2 < -2
Since the inequality is less than and not less than or equal to the
statement is false
The original statement is true
Let D = E = {−2,−1,0,1,2}.
d. ∃x in D such that ∀y in E, x ≤ y.
Determine which is true. Original or Negation?
~(∃x in D such that ∀y in E, x ≤ y)
Negation: ∀ x in D, ∃ y in E such that x >y
“For all x in D, there exists a y in E such that x>y”
Let x= 2 and y= 2
When we plug in x into the negation we get:
2>2
Since the inequality is less than and not less than or equal to
the statement is false
The original statement is true
Bibliography
• Charles Sanders Peirce- Biography. (n.d.). The European Graduate
School- Media and Cmmunication- Graduate and Postgraduate
Studies Program. Retrieved October 09, 2013 from
• http:??www.egs.edu/library/charles-sanders-peirce/bibliography