By: Savanah Yam, Rafael Diaz, William Miller 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 non-mathematical 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.”

(b) ∀ x, Real (x) ^ Neg(x) ->Pos(-x) (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/>.