Ch.1: Speaking Mathematically

Chapter 1 of the book "Discrete Mathematics with Applications" by Susanna S. Epp (4th edition) 1.1. Variables 1.2. The Language of Sets 1.3. The Language of Relations and Functions

66 flashcards · Mathematics · by lakesare

Part of Textbooks › Discrete Mathematics with Applications (Susanna Epp) · 12 courses

[Variables] What are the two uses of a variable in mathematics?
Placeholder for unknown values You imagine that the variable has one or more values, but you don't know what they are. Example: "Is there a number xxx with the property that 2x+3=x22x + 3 = x^22x+3=x2 ?" Arbitrary element of a set You want what you say about the variable to be equally true for all elements of a given set. Example: "No matter what number nnn might be chosen, if nnn is greater than 2, then n2n^2n2 is greater than 4." In both uses, the variable gives a temporary name to something…
[Variables] Rewrite with variables: "Are there numbers with the property that the sum of their squares equals the square of their sum?"
Are there numbers aaa and bbb such that a2+b2=(a+b)2a^2 + b^2 = (a + b)^2a2+b2=(a+b)2 ? The answer is yes: a=1a = 1a=1 , b=0b = 0b=0 gives 1+0=1=(1+0)21 + 0 = 1 = (1 + 0)^21+0=1=(1+0)2 .
[Variables] Rewrite with a variable: "Given any real number, its square is nonnegative."
For all real numbers rrr , r2≥0r^2 \ge 0r2≥0 . Or: Given any real number rrr , r2r^2r2 is nonnegative.
[Variables] What are three of the most important kinds of statements in mathematics?
Universal statement Says that a certain property is true for all elements in a set. Example: All positive numbers are greater than zero. Conditional statement Says that if one thing is true, then some other thing also has to be true. Example: If 378 is divisible by 18, then 378 is divisible by 6. Existential statement Given a property that may or may not be true, says that there is at least one thing for which the property is true. Example: There is a prime number that is even.
[Variables] What is a universal conditional statement?
A statement that is both universal ("for all") and conditional ("if-then"). Example: For all animals aaa , if aaa is a dog, then aaa is a mammal.
[Variables] For all animals aaa , if aaa is a dog, then aaa is a mammal.Rewrite this so that it looks purely conditional, and so that it looks purely universal.
Conditional: If aaa is a dog, then aaa is a mammal. Universal: All dogs are mammals. Or: "If an animal is a dog, then the animal is a mammal" and "For all dogs aaa , aaa is a mammal."
[Variables] For all real numbers xxx , if xxx is nonzero then x2x^2x2 is positive.Fill in the blanks to rewrite this: "If xxx ___, then ___."
If xxx is a nonzero real number, then x2x^2x2 is positive. A variable introduced in the first part of the sentence must also be used in the second part.Other forms: "For all nonzero real numbers xxx , x2x^2x2 is positive." "All nonzero real numbers have positive squares."
[Variables] What is a universal existential statement?
A statement that says that for all objects of a given type, something exists. The first part is universal, the second part existential.Example: Every real number has an additive inverse. (An additive inverse of aaa is a number bbb with a+b=0a + b = 0a+b=0 .)
[Variables] Rewrite with variables rrr and sss : "Every real number has an additive inverse."
For all real numbers rrr , there is a real number sss such that sss is an additive inverse for rrr . The additive inverse sss depends on rrr : when rrr is positive, sss is negative; when rrr is negative, sss is positive; when rrr is zero, sss is zero.
[Variables] Fill in the blank to rewrite "Every pot has a lid": "For all pots PPP , there is a lid LLL such that ___."
LLL is a lid for PPP . Less formally: "All pots have lids" or "For all pots PPP , there is a lid for PPP ."
[Variables] What is an existential universal statement?
A statement that asserts that some object exists and that this object has a property for all things of a certain kind. Example: There is a positive integer that is less than or equal to every positive integer. (True: the number 1.)
[Variables] Rewrite with variables mmm and nnn : "There is a positive integer that is less than or equal to every positive integer."
There is a positive integer mmm such that for all positive integers nnn , m≤nm \le nm≤n . The statement is true with m=1m = 1m=1 .
[Variables] Rewrite with variables ppp and qqq : "There is a person in my class who is at least as old as every person in my class."
There is a person ppp in my class such that for every person qqq in my class, ppp is at least as old as qqq .
[Variables] a1,a2,a3,…a_1, a_2, a_3, \dotsa1,a2,a3,… is a sequence of real numbers. What does "the limit of ana_nan as nnn approaches infinity is LLL " mean?
For all positive real numbers ε\varepsilonε , there is an integer NNN such that for all integers nnn , if n>Nn > Nn>N then −ε<an−L<ε-\varepsilon < a_n - L < \varepsilon−ε<an−L<ε . The definition needs all three phrases "for all", "there is" and "if-then".
[The Language of Sets] Who introduced the word set as a formal mathematical term, and when?
Georg Cantor, in 1879. Cantor thought of a set simply as a collection of elements.
[The Language of Sets] SSS is a set. What do x∈Sx \in Sx∈S and x∉Sx \notin Sx∈/S mean?
x∈Sx \in Sx∈S : xxx is an element of SSS . x∉S\ x \notin S x∈/S : xxx is not an element of SSS . Example: 57∈{1,2,3,…,100}57 \in \{1, 2, 3, \dots, 100\}57∈{1,2,3,…,100} and 0∉{1,2,3,…,100}0 \notin \{1, 2, 3, \dots, 100\}0∈/{1,2,3,…,100} .
[The Language of Sets] What is the set-roster notation?
Writing all elements of a set between braces. Example: {1,2,3}\{1, 2, 3\}{1,2,3} is the set whose elements are 1, 2 and 3.Example: {1,2,3,… }\{1, 2, 3, \dots\}{1,2,3,…} is the set of all positive integers.
[The Language of Sets] In {1,2,3,…,100}\{1, 2, 3, \dots, 100\}{1,2,3,…,100} the symbol "..." is called an ellipsis and is read "and so forth".
[The Language of Sets] What does the axiom of extension say?
A set is completely determined by its elements, not by the order in which they are listed or by how often an element is listed. Example: A={1,2,3}A = \{1, 2, 3\}A={1,2,3} , B={3,1,2}B = \{3, 1, 2\}B={3,1,2} and C={1,1,2,3,3,3}C = \{1, 1, 2, 3, 3, 3\}C={1,1,2,3,3,3} are the same set.
[The Language of Sets] Is {0}=0\{0\} = 0{0}=0 ?
No. {0}\{0\}{0} is a set with one element, namely 0, whereas 0 is the number zero.
[The Language of Sets] How many elements does the set {1,{1}}\{1, \{1\}\}{1,{1}} have?
Two: the number 1 and the set {1}\{1\}{1} whose only element is 1.
[The Language of Sets] For each nonnegative integer nnn , Un={n,−n}U_n = \{n, -n\}Un={n,−n} . What are U1U_1U1 and U0U_0U0 ?
U1={1,−1}\boldsymbol{U_1 = \{1, -1\}}U1={1,−1} , U0={0}\ \boldsymbol{U_0 = \{0\}} U0={0} U0={0,−0}={0,0}={0}U_0 = \{0, -0\} = \{0, 0\} = \{0\}U0={0,−0}={0,0}={0} , a set with one element.
[The Language of Sets] Which sets do the symbols R\mathbf{R}R , Z\mathbf{Z}Z and Q\mathbf{Q}Q denote?
R\mathbf{R}R The set of all real numbers. Z\mathbf{Z}Z The set of all integers. Z is the first letter of the German word for integers, Zahlen. Q\mathbf{Q}Q The set of all rational numbers, i.e. quotients of integers.
[The Language of Sets] What do R+\mathbf{R}^+R+ , Z−\mathbf{Z}^-Z− and Znonneg\mathbf{Z}^{\text{nonneg}}Znonneg denote?
The positive real numbers, the negative integers, and the nonnegative integers. A superscript +++ , −-− or nonneg keeps only the positive, negative or nonnegative elements of the set.Example: Znonneg={0,1,2,3,… }\mathbf{Z}^{\text{nonneg}} = \{0, 1, 2, 3, \dots\}Znonneg={0,1,2,3,…} .0 is neither positive nor negative.
[The Language of Sets] Why is the phrase natural numbers ambiguous?
Authors disagree on its meaning: some mean the nonnegative integers {0,1,2,… }\{0, 1, 2, \dots\}{0,1,2,…} , others only the positive integers {1,2,3,… }\{1, 2, 3, \dots\}{1,2,3,…} .
[The Language of Sets] Why is the real number line called continuous, and the set of integers discrete?
The real line is imagined to have no holes; the integers are points at fixed intervals, separated from each other. The name discrete mathematics comes from this distinction between continuous and discrete objects.
[The Language of Sets] What does the set-builder notation {x∈S∣P(x)}\{x \in S \mid P(x)\}{x∈S∣P(x)} denote, and how is it read?
The set of all elements xxx in SSS such that P(x)P(x)P(x) is true. Read: "the set of all xxx in SSS such that P(x)P(x)P(x) ". SSS is a set and P(x)P(x)P(x) is a property that elements of SSS may or may not satisfy.Example: {n∈Z∣0<n<4}={1,2,3}\{n \in \mathbf{Z} \mid 0 < n < 4\} = \{1, 2, 3\}{n∈Z∣0<n<4}={1,2,3} .
[The Language of Sets] Why must the notation {x∣P(x)}\{x \mid P(x)\}{x∣P(x)} , without a set SSS that xxx comes from, be used with care?
Unrestricted use of it can lead to genuine contradictions in set theory. It is safe to use as a shorthand when the set SSS could be specified if necessary.
[The Language of Sets] Describe the set {x∈R∣−2<x<5}\{x \in \mathbf{R} \mid -2 < x < 5\}{x∈R∣−2<x<5} .
The open interval of real numbers strictly between −2-2−2 and 555 . The endpoints −2-2−2 and 555 are not in the set.
[The Language of Sets] Write {x∈Z∣−2<x<5}\{x \in \mathbf{Z} \mid -2 < x < 5\}{x∈Z∣−2<x<5} and {x∈Z+∣−2<x<5}\{x \in \mathbf{Z}^+ \mid -2 < x < 5\}{x∈Z+∣−2<x<5} in set-roster notation.
{−1,0,1,2,3,4}\boldsymbol{\{-1, 0, 1, 2, 3, 4\}}{−1,0,1,2,3,4} and {1,2,3,4}\boldsymbol{\{1, 2, 3, 4\}}{1,2,3,4} Z+\mathbf{Z}^+Z+ contains only positive integers.
[The Language of Sets] When is AAA a subset of BBB , written A⊆BA \subseteq BA⊆B ?
When every element of AAA is also an element of BBB : for all elements xxx , if x∈Ax \in Ax∈A then x∈Bx \in Bx∈B . Also said: " AAA is contained in BBB " or " BBB contains AAA ".Example: {1,3}⊆{1,2,3}\{1, 3\} \subseteq \{1, 2, 3\}{1,3}⊆{1,2,3} .
[The Language of Sets] What does A⊈BA \not\subseteq BA⊆B mean?
There is at least one element xxx such that x∈Ax \in Ax∈A and x∉Bx \notin Bx∈/B . Example: {1,4}⊈{1,2,3}\{1, 4\} \not\subseteq \{1, 2, 3\}{1,4}⊆{1,2,3} because of 4.
[The Language of Sets] When is AAA a proper subset of BBB ?
When every element of AAA is in BBB , but at least one element of BBB is not in AAA . Example: {1,2}\{1, 2\}{1,2} is a proper subset of {1,2,3}\{1, 2, 3\}{1,2,3} ; {1,2,3}\{1, 2, 3\}{1,2,3} is a subset of itself but not a proper subset.
[The Language of Sets] Is every set CCC a subset of itself?
Yes: every element of CCC is in CCC , so C⊆CC \subseteq CC⊆C .
[The Language of Sets] B={n∈Z∣0≤n≤100}B = \{n \in \mathbf{Z} \mid 0 \le n \le 100\}B={n∈Z∣0≤n≤100} . Is B⊆Z+B \subseteq \mathbf{Z}^+B⊆Z+ ?
No: 0∈B0 \in B0∈B but 0 is not a positive integer.
[The Language of Sets] C={100,200,300,400,500}C = \{100, 200, 300, 400, 500\}C={100,200,300,400,500} and B={n∈Z∣0≤n≤100}B = \{n \in \mathbf{Z} \mid 0 \le n \le 100\}B={n∈Z∣0≤n≤100} . Is C⊆BC \subseteq BC⊆B ? Is CCC a proper subset of Z+\mathbf{Z}^+Z+ ?
C⊈BC \not\subseteq BC⊆B ; CCC is a proper subset of Z+\mathbf{Z}^+Z+ . 200∈C200 \in C200∈C but 200∉B200 \notin B200∈/B . Every element of CCC is a positive integer, and 1∈Z+1 \in \mathbf{Z}^+1∈Z+ is not in CCC .
[The Language of Sets] Which of these are true? (a) 2∈{1,2,3}2 \in \{1, 2, 3\}2∈{1,2,3} \quad (b) {2}∈{1,2,3}\{2\} \in \{1, 2, 3\}{2}∈{1,2,3} \quad (c) 2⊆{1,2,3}2 \subseteq \{1, 2, 3\}2⊆{1,2,3} (d) {2}⊆{1,2,3}\{2\} \subseteq \{1, 2, 3\}{2}⊆{1,2,3} \quad (e) {2}⊆{{1},{2}}\{2\} \subseteq \{\{1\}, \{2\}\}{2}⊆{{1},{2}} \quad (f) {2}∈{{1},{2}}\{2\} \in \{\{1\}, \{2\}\}{2}∈{{1},{2}}
Only (a), (d) and (f). (b): the elements of {1,2,3}\{1, 2, 3\}{1,2,3} are 1, 2, 3, and 2≠{2}2 \ne \{2\}2={2} . (c): the number 2 is not a set.(e): the element 2 of {2}\{2\}{2} is equal to neither {1}\{1\}{1} nor {2}\{2\}{2} .
[The Language of Sets] What problem had to be solved to define an ordered pair using only sets?
A set does not depend on the order of its elements ( {a,b}={b,a}\{a, b\} = \{b, a\}{a,b}={b,a} ), but an ordered pair must say which element comes first. Norbert Wiener and Felix Hausdorff gave awkward definitions in 1914; Kazimierz Kuratowski gave the standard one in 1921.
[The Language of Sets] What set is the ordered pair (a,b)(a, b)(a,b) in Kuratowski's definition (1921)?
{{a},{a,b}}\boldsymbol{\{\{a\}, \{a, b\}\}}{{a},{a,b}} If a≠ba \ne ba=b , then aaa is in both elements {a}\{a\}{a} and {a,b}\{a, b\}{a,b} , and bbb only in {a,b}\{a, b\}{a,b} , so aaa is the first element and bbb the second.If a=ba = ba=b , the set is {{a},{a,a}}={{a}}\{\{a\}, \{a, a\}\} = \{\{a\}\}{{a},{a,a}}={{a}} , and aaa is both the first and the second element.
[The Language of Sets] When are two ordered pairs (a,b)(a, b)(a,b) and (c,d)(c, d)(c,d) equal?
If, and only if, a=ca = ca=c and b=db = db=d . Example: (1,2)≠(2,1)(1, 2) \ne (2, 1)(1,2)=(2,1) since 1≠21 \ne 21=2 .Example: (3,510)=(9,12)\left(3, \frac{5}{10}\right) = \left(\sqrt{9}, \frac{1}{2}\right)(3,105)=(9,21) since 3=93 = \sqrt{9}3=9 and 510=12\frac{5}{10} = \frac{1}{2}105=21 .
[The Language of Sets] What is the Cartesian product A×BA \times BA×B of sets AAA and BBB ?
The set of all ordered pairs (a,b)(a, b)(a,b) with a∈Aa \in Aa∈A and b∈Bb \in Bb∈B : A×B={(a,b)∣a∈A and b∈B}\ A \times B = \{(a, b) \mid a \in A \text{ and } b \in B\} A×B={(a,b)∣a∈A and b∈B} . Read " AAA cross BBB ".Example: {1,2}×{u}={(1,u),(2,u)}\{1, 2\} \times \{u\} = \{(1, u), (2, u)\}{1,2}×{u}={(1,u),(2,u)} .
[The Language of Sets] A={1,2,3}A = \{1, 2, 3\}A={1,2,3} and B={u,v}B = \{u, v\}B={u,v} . Find A×BA \times BA×B and B×BB \times BB×B .
A×B={(1,u),(2,u),(3,u),(1,v),(2,v),(3,v)}\boldsymbol{A \times B = \{(1, u), (2, u), (3, u), (1, v), (2, v), (3, v)\}}A×B={(1,u),(2,u),(3,u),(1,v),(2,v),(3,v)} , B×B={(u,u),(u,v),(v,u),(v,v)}\ \boldsymbol{B \times B = \{(u, u), (u, v), (v, u), (v, v)\}} B×B={(u,u),(u,v),(v,u),(v,v)} B×A={(u,1),(u,2),(u,3),(v,1),(v,2),(v,3)}B \times A = \{(u, 1), (u, 2), (u, 3), (v, 1), (v, 2), (v, 3)\}B×A={(u,1),(u,2),(u,3),(v,1),(v,2),(v,3)} is a different set: its pairs have the letter first.
[The Language of Sets] AAA has mmm elements and BBB has nnn elements. How many elements does A×BA \times BA×B have?
m⋅n\boldsymbol{m \cdot n}m⋅n This is why the Cartesian product is called a product.Example: {1,2,3}×{u,v}\{1, 2, 3\} \times \{u, v\}{1,2,3}×{u,v} has 3⋅2=63 \cdot 2 = 63⋅2=6 elements.
[The Language of Sets] What is R×R\mathbf{R} \times \mathbf{R}R×R , and how is it pictured?
The set of all ordered pairs (x,y)(x, y)(x,y) of real numbers, pictured as the points of the Cartesian plane. The first element of the pair gives the horizontal position of the point, the second the vertical position.
[The Language of Relations and Functions] A={0,1,2}A = \{0, 1, 2\}A={0,1,2} , B={1,2,3}B = \{1, 2, 3\}B={1,2,3} , and x∈Ax \in Ax∈A is related to y∈By \in By∈B if, and only if, x<yx < yx<y . Which ordered pairs in A×BA \times BA×B have related elements?
(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)\boldsymbol{(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)}(0,1),(0,2),(0,3),(1,2),(1,3),(2,3) Knowing this set of pairs is the same as knowing which elements are related to which. So the relation can be defined as this set of ordered pairs.
[The Language of Relations and Functions] What is a relation RRR from a set AAA to a set BBB ?
A subset of A×BA \times BA×B . xxx is related to yyy by RRR , written xRyx \mathrel{R} yxRy , if, and only if, (x,y)∈R(x, y) \in R(x,y)∈R . AAA is the domain of RRR , BBB its co-domain.This definition was introduced by C. S. Peirce in the nineteenth century.Example: R={(1,2)}R = \{(1, 2)\}R={(1,2)} is a relation from {1}\{1\}{1} to {2,3}\{2, 3\}{2,3} , with 1R21 \mathrel{R} 21R2 .
[The Language of Relations and Functions] RRR is a relation from AAA to BBB . What does xR̸yx \not\mathrel{R} yxRy mean?
(x,y)∉R(x, y) \notin R(x,y)∈/R , i.e. xxx is not related to yyy by RRR .
[The Language of Relations and Functions] A={1,2}A = \{1, 2\}A={1,2} , B={1,2,3}B = \{1, 2, 3\}B={1,2,3} , and the relation RRR from AAA to BBB is: (x,y)∈R(x, y) \in R(x,y)∈R means that x−y2\frac{x - y}{2}2x−y is an integer. Write RRR as a set of ordered pairs.
R={(1,1),(1,3),(2,2)}\boldsymbol{R = \{(1, 1), (1, 3), (2, 2)\}}R={(1,1),(1,3),(2,2)} 1−12=0\frac{1-1}{2} = 021−1=0 , 1−32=−1\frac{1-3}{2} = -121−3=−1 , 2−22=0\frac{2-2}{2} = 022−2=0 are integers; 1−22=−12\frac{1-2}{2} = -\frac{1}{2}21−2=−21 , 2−12=12\frac{2-1}{2} = \frac{1}{2}22−1=21 , 2−32=−12\frac{2-3}{2} = -\frac{1}{2}22−3=−21 are not.So 1R31 \mathrel{R} 31R3 but 2R̸32 \not\mathrel{R} 32R3 .
[The Language of Relations and Functions] The circle relation CCC from R\mathbf{R}R to R\mathbf{R}R : (x,y)∈C(x, y) \in C(x,y)∈C means that x2+y2=1x^2 + y^2 = 1x2+y2=1 . Which are in CCC : (1,0)(1, 0)(1,0) , (0,0)(0, 0)(0,0) , (−12,32)\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)(−21,23) , (−2,0)(-2, 0)(−2,0) , (0,−1)(0, -1)(0,−1) , (1,1)(1, 1)(1,1) ?
Not in CCC : (0,0)(0, 0)(0,0) , (−2,0)(-2, 0)(−2,0) , (1,1)(1, 1)(1,1) . The other three are in CCC . 14+34=1\frac{1}{4} + \frac{3}{4} = 141+43=1 , but 0+0=00 + 0 = 00+0=0 , 4+0=44 + 0 = 44+0=4 , 1+1=21 + 1 = 21+1=2 . Plotted, CCC is the circle of radius 1 around the origin.
[The Language of Relations and Functions] How is the arrow diagram of a relation RRR from AAA to BBB drawn?
Draw the elements of AAA as points in one region and those of BBB in another; draw an arrow from xxx to yyy if, and only if, xRyx \mathrel{R} yxRy . Example: A={1,2,3}A = \{1, 2, 3\}A={1,2,3} , B={1,3,5}B = \{1, 3, 5\}B={1,3,5} , (x,y)∈S(x, y) \in S(x,y)∈S means x<yx < yx<y , and T={(2,1),(2,5)}T = \{(2, 1), (2, 5)\}T={(2,1),(2,5)} : An element of AAA can have several arrows coming out of it, or none.
[The Language of Relations and Functions] What two properties make a relation FFF from AAA to BBB a function from AAA to BBB ?
At least one For every element xxx in AAA , there is an element yyy in BBB such that (x,y)∈F(x, y) \in F(x,y)∈F . Example: {(1,a)}\{(1, a)\}{(1,a)} is not a function from {1,2}\{1, 2\}{1,2} to {a,b}\{a, b\}{a,b} : no pair has first element 2. At most one For all elements xxx in AAA and yyy and zzz in BBB , if (x,y)∈F(x, y) \in F(x,y)∈F and (x,z)∈F(x, z) \in F(x,z)∈F , then y=zy = zy=z . Example: {(1,a),(1,b),(2,a)}\{(1, a), (1, b), (2, a)\}{(1,a),(1,b),(2,a)} is not a function from {1,2}\{1, 2\…
[The Language of Relations and Functions] State the two properties of a function FFF from AAA to BBB in terms of its ordered pairs.
Every element of AAA is the first element of an ordered pair of FFF , and no two distinct ordered pairs of FFF have the same first element. Example: {(1,a),(2,a)}\{(1, a), (2, a)\}{(1,a),(2,a)} is a function from {1,2}\{1, 2\}{1,2} to {a,b}\{a, b\}{a,b} .
[The Language of Relations and Functions] State the two properties of a function in terms of its arrow diagram.
There is an arrow coming out of each element of the domain, and no element of the domain has more than one arrow coming out of it. A function from {2,4,6}\{2, 4, 6\}{2,4,6} to {1,3,5}\{1, 3, 5\}{1,3,5} .
[The Language of Relations and Functions] FFF is a function from AAA to BBB and x∈Ax \in Ax∈A . What does F(x)F(x)F(x) denote?
The unique element of BBB that is related to xxx by FFF . Read " FFF of xxx ". It exists because every x∈Ax \in Ax∈A is related to some element of BBB , and it is unique because no xxx is related to two different elements of BBB .
[The Language of Relations and Functions] A={2,4,6}A = \{2, 4, 6\}A={2,4,6} , B={1,3,5}B = \{1, 3, 5\}B={1,3,5} . Is R={(2,5),(4,1),(4,3),(6,5)}R = \{(2, 5), (4, 1), (4, 3), (6, 5)\}R={(2,5),(4,1),(4,3),(6,5)} a function from AAA to BBB ?
No: 4 is related to two elements, 1 and 3. Two arrows come out of 4.
[The Language of Relations and Functions] A={2,4,6}A = \{2, 4, 6\}A={2,4,6} , B={1,3,5}B = \{1, 3, 5\}B={1,3,5} , and (x,y)∈S(x, y) \in S(x,y)∈S means that y=x+1y = x + 1y=x+1 . Is SSS a function from AAA to BBB ?
No: 6 is related to no element of BBB , since 6+1=7∉B6 + 1 = 7 \notin B6+1=7∈/B . S={(2,3),(4,5)}S = \{(2, 3), (4, 5)\}S={(2,3),(4,5)} ; no arrow comes out of 6.
[The Language of Relations and Functions] TTT is the relation from {2,4,6}\{2, 4, 6\}{2,4,6} to {1,3,5}\{1, 3, 5\}{1,3,5} with this arrow diagram. Is TTT a function? If so, find T(2)T(2)T(2) , T(4)T(4)T(4) , T(6)T(6)T(6) .
Yes: T(2)=5\boldsymbol{T(2) = 5}T(2)=5 , T(4)=1\boldsymbol{T(4) = 1}T(4)=1 , T(6)=1\boldsymbol{T(6) = 1}T(6)=1 . Exactly one arrow comes out of each element of {2,4,6}\{2, 4, 6\}{2,4,6} .
[The Language of Relations and Functions] Can two different elements of the domain of a function be related to the same element of the co-domain?
Yes. The definition only forbids one element of the domain to be related to two different elements of the co-domain. Example: the function T={(2,5),(4,1),(6,1)}T = \{(2, 5), (4, 1), (6, 1)\}T={(2,5),(4,1),(6,1)} from {2,4,6}\{2, 4, 6\}{2,4,6} to {1,3,5}\{1, 3, 5\}{1,3,5} has T(4)=T(6)=1T(4) = T(6) = 1T(4)=T(6)=1 .
[The Language of Relations and Functions] The circle relation CCC from R\mathbf{R}R to R\mathbf{R}R : (x,y)∈C(x, y) \in C(x,y)∈C means x2+y2=1x^2 + y^2 = 1x2+y2=1 . Is CCC a function?
No, both function properties fail. x=2x = 2x=2 is related to no yyy : 4+y2=14 + y^2 = 14+y2=1 would need y2=−3y^2 = -3y2=−3 . x=12\ x = \frac{1}{2} x=21 is related to two values, y=±32y = \pm\frac{\sqrt{3}}{2}y=±23 . The vertical line x=12x = \frac{1}{2}x=21 meets the graph of CCC twice.
[The Language of Relations and Functions] The relation LLL from R\mathbf{R}R to R\mathbf{R}R : (x,y)∈L(x, y) \in L(x,y)∈L means y=x−1y = x - 1y=x−1 . Is LLL a function? If so, find L(0)L(0)L(0) and L(1)L(1)L(1) .
Yes: L(0)=−1\boldsymbol{L(0) = -1}L(0)=−1 , L(1)=0\boldsymbol{L(1) = 0}L(1)=0 . For each real xxx , y=x−1y = x - 1y=x−1 is a real number; and if (x,y),(x,z)∈L(x, y), (x, z) \in L(x,y),(x,z)∈L , then y=x−1=zy = x - 1 = zy=x−1=z . For every real xxx , the vertical line through (x,0)(x, 0)(x,0) meets the graph of LLL exactly once.
[The Language of Relations and Functions] What is a function machine?
A picture of a function fff from XXX to YYY as a machine that takes an input x∈Xx \in Xx∈X and produces the output f(x)f(x)f(x) .
[The Language of Relations and Functions] What are the squaring function, the successor function and a constant function?
Squaring function fff from R\mathbf{R}R to R\mathbf{R}R with f(x)=x2f(x) = x^2f(x)=x2 , i.e. f(□)=□2f(\square) = \square^2f(□)=□2 , or f:x→x2f: x \to x^2f:x→x2 . Example: f(−3)=9f(-3) = 9f(−3)=9 . Successor function ggg from Z\mathbf{Z}Z to Z\mathbf{Z}Z with g(n)=n+1g(n) = n + 1g(n)=n+1 , i.e. g(□)=□+1g(\square) = \square + 1g(□)=□+1 , or g:n→n+1g: n \to n + 1g:n→n+1 . Example: g(−1000)=−999g(-1000) = -999g(−1000)=−999 . Constant function A function whose output is the same for every input, e.g…
[The Language of Relations and Functions] In " f(x)=x2f(x) = x^2f(x)=x2 for all real numbers xxx ", the xxx is a dummy variable: any other symbol could replace it, as long as it is replaced everywhere.
Example: f(t)=t2f(t) = t^2f(t)=t2 for all real ttt defines the same function.
[The Language of Relations and Functions] fff is a function from AAA to BBB . Which subset of A×BA \times BA×B is fff ?
f={(x,y)∈A×B∣y=f(x)}\boldsymbol{f = \{(x, y) \in A \times B \mid y = f(x)\}}f={(x,y)∈A×B∣y=f(x)} A function is a special kind of relation, and a relation is a subset of a Cartesian product.Example: f:{−1,0,1}→{0,1}f: \{-1, 0, 1\} \to \{0, 1\}f:{−1,0,1}→{0,1} with f(x)=x2f(x) = x^2f(x)=x2 is the set {(−1,1),(0,0),(1,1)}\{(-1, 1), (0, 0), (1, 1)\}{(−1,1),(0,0),(1,1)} .
[The Language of Relations and Functions] fff and ggg are functions from AAA to BBB . When is f=gf = gf=g ?
If, and only if, f(x)=g(x)f(x) = g(x)f(x)=g(x) for all xxx in AAA . Both are sets of ordered pairs (x,f(x))(x, f(x))(x,f(x)) and (x,g(x))(x, g(x))(x,g(x)) , so they are equal exactly when the pairs agree.
[The Language of Relations and Functions] f,g:R→Rf, g: \mathbf{R} \to \mathbf{R}f,g:R→R are defined by f(x)=∣x∣f(x) = |x|f(x)=∣x∣ and g(x)=x2g(x) = \sqrt{x^2}g(x)=x2 . Is f=gf = gf=g ?
Yes: for every real number xxx , the absolute value ∣x∣|x|∣x∣ equals the square root of x2x^2x2 . Example: f(−3)=3=9=g(−3)f(-3) = 3 = \sqrt{9} = g(-3)f(−3)=3=9=g(−3) .