Project 1
By: Savanah Yam, Rafael Diaz,
William Miller
Biography
• Mathematician, German Gottlob Frege, first introduced predicates
and quantifiers that are accepted today. Quantifiers and predicates
are a formalized version of proofs. Frege showed society that
people could use his system to decipher the theoretical math
statements into simplest form with mathematical approach and
notations. He led out of the mathematical approach so that
predicates and quantifiers can be represented in a nonmathematical thoughts and predictions. This then led to symbols to
signify long sentences into shorter ones involving symbols.
Generally, these sentences are complex. To express these
statements, he used various notations: statement, negation,
conditional and generality. Because Frege took both a mathematical
and non-mathematical approach the logic can be applicable to
anything. Frege was very precise and accurate with his description
and definition of logical mathematical concepts.
Question #28 Sec 3.1
• Let the domain of x be the set D of objects
discussed in mathematics courses, and let
Real(x) be “x is a real number,” Pos(x) be “x is
a positive real number”,. Neg(x) be “x is a
negative number,” and Int(x) be “x is an
integer.”
• (a) Pos(0)
– 0 is a positive real number.
– This is false because 0 is not a positive number.
• (b) ∀ x, Real (x) ^ Neg(x) ->Pos(-x)
– If a real number is negative, then its appositive
will be a positive real number.
– True, because for all real numbers x, -(-|x|)=|x|
• (c) ∀ x, Int(x)->Real(x)
– If x is an integer , then x is a real number.
– True, every integer is a real number.
• Ex such that Real(x)^~Int(x)
– There is a real number that is not an integer.
– True, fractions like 2/3 is a real number that is not
an integer.
Question #29 Sec 3.1
• Let the domain of x be the set of geometric
figures in the plane and let Square(x) be “x is a
square” and Rect(x) be “x is a rectangle”.
• (a)Given: ∃ x such that Rect(x)^Square(x)
– True, since a square is a rectangle,
Rect(x)^Square(x)=Square(x)
• (b)Given: ∃ x such that Rect(x)^~Square(x)
– Consider the rectangle x of length 2 cm and width
1 cm, therefore, x is not a square.
– Then
• x ∈ Rect(x), X ∉ Square(x)
• X ∈ Rect (x), x ∈ ~Square(x)
• X ∈ Rect(x)^~Square(x)
– Therefore the statement is true.
• (c)Given: ∀x, Square(x)->Rect(x)
– The statement is true. Since a square is a
rectangle.
Question #30 Sec 3.1
• Let the domain of x be the set Z of integers,
and let Odd(x) be “x is odd”, Prime(x) be “x is
prime”, and Square(x) be “x is a perfect
square.”
• (a) ∃x such that prime (x)^~ odd(x)
– There is a prime number that is not an odd
integer.
– This is true because 2 is a prime integer but not an
odd integer.
• (b) ∀x, prime(x)->~square(x)
– If an integer is prime, then it is not a perfect
square.
– This statement is true because a prime number is
an integer greater than 1 that is not a product of
two smaller positive integers. So a prime number
cannot be a perfect square, because it would have
to be a product of two smaller positive integers.
• (c) ∃x such that odd (x)^square(x)
– There is an odd integer that is a perfect square.
– True because 25 is an odd integer and 25=5^2
Question #20 Sec 3.3
• Recall that reversing the order of the
quantifiers in a statement with two different
quantifiers may change the truth value of the
statement, but it does not necessarily do so.
All of the statements on the next page refer to
the Tarski world in figure 3.3.1. In each pair,
the order of the quantifiers is reversed but
everything else is the same. For each pair,
determine whether the statements have the
same or opposite truth values.
• (a)Given:
– (1) For all squares y there is a triangle x, such that
x and y
– (2) There is a triangle x such that for all squares y,
x and y have different colors.
– Statement (1) is true.
• From the picture {e,g,h,j} are squares and {d,f,i} are
triangles. For and square y we can find a triangle x such
that x,y are of a different color.
– Statement (2) is true.
• From the diagram if we fix the triangle “I” for x, then for
any square y we have x, y are of different colors.
• Given:
– (1) for all circles y there is a square that x and y
have the same color.
– (2) There is a square x, such that for all circles y, x
and y have the same color.
– Statement (1) is false
• There is not a square x such that x and y have the same
color
– Statement (2) is false
• If we fix square “j” for x then j and b are of different
color.
Bibliography
• "Mathematical Symbols:
EnchantedLearning.com." Mathematical Symbols:
EnchantedLearning.com. N.p., n.d. Web. 15 Mar. 2014.
<http://www.enchantedlearning.com/math/s
• "Mathwords." Mathwords. N.p., n.d. Web. 15 Mar.
2014. <http://www.mathwords.com/>.
• Zalta, Edward N. "Gottlob Frege." Stanford University.
14 Sept. 1995. Stanford University. 13 Mar. 2014
<http://plato.stanford.edu/entries/frege/>.