Chapter 1

set-builder notation

A notation used to describe the elements of a set

ordered pair

A pair of numbers that can be used to locate a point on a coordinate plane

Universal Existential Statement

A statement that is universal because its first part says that a certain property is true for all objects of a given type, and it is existential because its second part asserts the existence of something

proper subset

A subset that does not contain every element in another set.

Let A = {2,4,6} and B = {1,3,5}. Which of the relations R, S, and T defined below are functions from A to B? a. R = {(2,5), (4,1), (4,3), (6,5)} b. For all (x,y) ∈ A×B, (x,y) ∈ S means that y=x+1. c. T is defined by the arrow diagram: (2,5) (4,1) (4,3) (6,5)

A. R is not a function because it does not satisfy property (2). The ordered pairs (4, 1) and (4, 3) have the same first element but different second elements b. S is not a function because it does not satisfy property (1). It is not true that every element of A is the first element of an ordered pair in S c. T is a function: Each element in {2, 4, 6} is related to some element in {1, 3, 5}

subset

a set that is part of a larger set

Existential Universal Statement

a statement that is existential because its first part asserts that a certain object exists and is universal because its second part says that the object satisfies a certain property for all things of a certain kind

which is true? a. 2 ∈ {1,2,3} b. {2} ∈ {1,2,3} d. {2} ⊆ {1,2,3} e. {2} ⊆ {{1},{2}} c. 2 ⊆ {1,2,3} f. {2} ∈ {{1},{2}}

a, d, f are true

Let A = {1,2,3} and B = {u,v}. a. Find A × B b. Find B×A c. Find B×B d. How many elements are in A×B, B×A, and B×B? e. Let R denote the set of all real numbers. Describe R × R.

a. A × B = {(1,u),(2,u),(3,u),(1,v),(2,v),(3,v)} b. B × A = {(u,1),(u,2),(u,3),(v,1),(v,2),(v,3)} c. B × B = {(u,u),(u,v),(v,u),(v,v)} d. A × B has six elements. B × A has six elements. B × B has four elements. e. R × R is the set of all ordered pairs (x, y) where both x and y are real numbers. If horizontal and vertical axes are drawn on a plane and a unit length is marked off, then each ordered pair in R × R corresponds to a unique point in the plane, with the first and second elements of the pair indicating, respectively, the horizontal and vertical positions of the point.

a. Let A = {1,2,3}, B = {3,1,2}, and C = {1,1,2,3,3,3}. What are the elements of A, B, and C? How are A, B, and C related? b. Is{0}=0? c. How many elements are in the set {1, {1}}? d. For each nonnegative integer n, let Un = {n, −n}. Find U1, U2, and U0.

a. A, B, and C have exactly the same three elements: 1, 2, and 3. Therefore, A, B, and C are simply different ways to represent the same set. b. {0} not = to 0 because {0} is a set with one element, namely 0, whereas 0 is just the symbol that represents the number zero. c. The set {1, {1}} has two elements: 1 and the set whose only element is 1. d. U1 = {1,−1}, U2 = {2,−2}, U0 = {0,−0} = {0,0} = {0}.

Let A = {1,2} and B = {1,2,3} and define a relation R from A to B as follows: Given any (x, y) ∈ A × B, (x,y)∈R means that (x−y)/2 is an integer. a. State explicitly which ordered pairs are in A × B and which are in R. b. Is 1 R 3? Is 2 R 3? Is 2 R 2? c. What are the domain and co-domain of R?

a. A×B={(1,1),(1,2),(1,3),(2,1),(2,2),(2,3)} b. Yes, 1 R 3 because(1,3)∈R. No, 2 R 3 because (2,3) ∈/ R. Yes, 2 R 2 because (2,2) ∈ R. C. The domain of R is {1, 2} and the co-domain is {1, 2, 3}.

Let A = Z+, B = {n ∈ Z|0 ≤ n ≤ 100}, and C = {100,200,300,400,500}. Evaluate the truth and falsity of each of the following statements. a. B ⊆ A b. C is a proper subset of A c. C and B have at least one element in common d. C ⊆ B e. C ⊆ C

a. False. Zero is not a positive integer. Thus zero is in B but zero is not in A, and so B A. b. True. Each element in C is a positive integer and, hence, is in A, but there are elements in A that are not in C. For instance, 1 is in A and not in C. c. True. For example, 100 is in both C and B. d. False. For example, 200 is in C but not in B. e. True. Every element in C is in C. In general, the definition of subset implies that all sets are subsets of themselves.

a. Is (1,2) = (2,1)? b.Is (3,5/10) = (square root of 9, 1/2)? c. What is the first element of (1, 1)?

a. No. b. Yes. c. In the ordered pair (1, 1), the first and the second elements are both 1.

Define a relation C from R to R as follows: For any (x,y) ∈ R×R, (x,y) ∈ C means that x^2 + y^2 = 1. a. Is (1,0) ∈ C? Is (0,0) ∈C ? Is (−1/2, √3/2) ∈ C? Is −2 C 0? Is 0 C (−1)? Is 1 C 1? b. What are the domain and co-domain of C? c. Draw a graph for C by plotting the points of C in the Cartesian plane.

a. Yes No Yes No Yes No b. The domain and co-domain of C are both R, the set of all real numbers. C. plot: (-1,0) (1,0) (0,1) (0,-1)

(Universal Existential Statement) Fill in the blanks to rewrite the following statement: Every pot has a lid. a. Allpots ____. b. For all pots P, there is _________ . c. Forallpots P,thereisalid L such that _________ .

a. have lids b. a lid for P c. L is a lid for P

(Universal Conditional Statement) For all real numbers x, if x is nonzero then x^2 is positive. a. If a real number is nonzero, then its square b. For all nonzero real numbers x,________. c. If x , then______. d. The square of any nonzero real number is e. All nonzero real numbers have ________. e. All nonzero real numbers have _______ .

a. is positive b. x^2 ispositive c. is a nonzero real number; x2 is positive d. positive e. positive squares

a. In Example 1.3.2 the circle relation C was defined as follows: For all (x,y) ∈ R×R, (x,y) ∈ C means that Is C a function? If it is, find C (0) and C (1). b. Define a relation from R to R as follows: x^2 +y^2 =1. For all (x,y) ∈ R×R, (x,y) ∈ L means that y=x−1. Is L a function? If it is, find L(0) and L(1).

a. no b. yes; L(0) = -1 ; L(1) = 0

(existential universal statement) There is a person in my class who is at least as old as every person in my class. a. Some _____ is atleast as old as _______ . b. There is person p in my class such that p is ________ . c. There is a person p in my class with the property that for every person q in my class, p is ________ .

a. person in my class; every person in my class b. at least as old as every person in my class c. atleast as old as q

Given that R denotes the set of all real numbers, Z the set of all integers, and Z+ the set of all positive integers, describe each of the following sets. a. {x∈R|−2<x<5} b. {x∈Z|−2<x<5} c. {x∈Z+ |−2<x<5}

a. {x ∈ R | −2 < x < 5} is the open interval of real numbers (strictly) between −2 and 5. b. {x ∈ Z |−2 < x < 5} is the set of all integers (strictly) between −2 and 5. It is equal to the set {−1,0,1,2,3,4}. c. Since all the integers in Z+ are positive, {x ∈ Z+|−2 < x < 5} = {1,2,3,4}.

Cartesian product of A and B

denoted A × B and read "A cross B," is the set of all ordered pairs (a,b), where a is in A and b is in B. Symbolically: A × B = {(a, b) | a ∈ A and b ∈ B} .

A function F from a set A to a set B

is a relation with domain A and co-domain B that satisfies the following two properties: 1: For every element x in A, there is an element y in B such that (x, y) ∈ F. 2. For all elements x in A and y and z in B, if (x,y) ∈ F and (x,z) ∈ F, then y = z.

Universal Conditional Statement

is a statement that is both universal and conditional.

Relation R from A to B

is a subset of A × B. Given an ordered pair (x, y) in A×B, x is related to y by R, written x R y, if, and only if, (x, y) is in R. The set A is called the domain of R and the set B is called its co-domain.

universal statement

says that a certain property is true for all elements in a set

Axiom of Extension

says that a set is completely determined by what its elements are—not the order in which they might be listed or the fact that some elements might be listed more than once.

conditional statement

says that if one thing is true then some other thing also has to be true.

existential statement

says that there is at least one thing for which the property is true

Ellipsis

three periods (...) indicating the omission of words in a thought or quotation

Define f: R→R and g:R→R by the following formulas: f(x)=|x| for all x ∈ R. g(x) =√x^2 for all x ∈ R. Does f = g?

yes