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Sets We won’t spend much time on the material from this and the next two chapters, Functions and Inverse Functions. That’s because these three chapters are mostly a review of some of the math that’s a prerequisite for this course. You don’t need to know all of it, but you should know much of it. Identify the topics in these three chapters that you are uncomfortable with, and spend the time to catch up on those topics. Ask your classmates, instructor, or perhaps a tutor to help you out. Sets A set is a collection of objects. For example, the set of countries in North America is a set that contains 3 objects: Canada, Mexico, and the United States. Set notation. Writing {2, 3, 5} is a shorthand for the set that contains the numbers 2, 3, and 5, and no objects other than 2, 3, and 5. The order in which the objects of a set are written doesn’t matter. For example, {5, 2, 3} and {2, 3, 5} are the same set. Alternatively, the previous sentence could be written as “For example, {5, 2, 3} = {2, 3, 5}.” If B is a set, and x is an object contained in B, we write x 2 B. If both x and y are objects contained in B then we write x, y 2 B. If x is not contained in B then we write x 2 / B. Examples. • 5 2 {2, 3, 5} • 2, 3 2 {2, 3, 5} •12 / {2, 3, 5} Subsets. One set is a subset of another set if every object in the first set is an object of the second set as well. The set of weekdays is a subset of the set of days of the week, since every weekday is a day of the week. A more succinct way to express the concept of a subset is as follows: The set B is a subset of the set C if every b 2 B is also contained in C. 11 Writing B ✓ C is a shorthand for writing “B is a subset of C ”. Writing B * C is a shorthand for writing “B is not a subset of C ”. Examples. • {2, 3} ✓ {2, 3, 5} • {2, 3, 5} * {3, 5, 7} Set minus. If A and B are sets, we can create a new set named A B (spoken as “A minus B”) by starting with the set A and removing all of the objects from A that are also contained in the set B. Examples. • {1, 7, 8} {7} = {1, 8} • {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} * * * * * {2, 4, 6, 8, 10} = {1, 3, 5, 7, 9} * * * * * * * * Numbers Among the most common sets appearing in math are sets of numbers. There are many di↵erent kinds of numbers. Below is a list of some that are important for this course. Natural numbers. N = {1, 2, 3, 4, ...} Integers. Z = {..., 2, 1, 0, 1, 2, 3, ...} Rational numbers. Q is the set of fractions of integers. That is, the n numbers contained in Q are exactly those of the form m where n and m are integers and m 6= 0. For example, 13 2 Q and 127 2 Q. Real numbers. R is the set of numbers that can be used to measure a distance or magnitude, or the negative of a number used to measure a distance 12 Rational thesetsetofoffractions fractions integers. is, the Rational numbers. numbers. QQ isis the of of integers. ThatThat is, the n numbers in QQ are areexactly exactlythose thoseof of form where n and m are numbers contained contained in thethe form n and m are m where integers integers and and m in ⌅= 01 0.0. For Q and and ~127E⇤Q~Q. For example, example, ~3 ⇤ E Q or magnitude. magnitude. The The set set ofof real real numbers numbers can can be be drawn drawn as as aa line line called called “the “the or number line”. number line”. Real numbers. numbers. JR cancan be be usedused to measure a Real R isis the the set setofofnumbers numbersthat that to measure a distance, or the negative of a number used to measure a distance. The set of distance, or the negative of a number used to measure a distance. The set of realnumbers numbers can can be number line”. real be drawn drawnasasaaline linecalled called“the “the number line”. p S S II) S i.W 2 lIT ⇧~ and ir are two of very many real numbers that are not rational numbers. 2 and are two of very many real numbers that are not rational numbers. (Aside: the definition of JR above isn’t very precise, and thus isn’t a very good (Aside: R above isn’t precise, and thus very good p the definition p definition. The set of of real numbers hasvery a better definition, butisn’t it’s aoutside 2and and⇡of ⇡The are two ofvery very many realhas numbers that are with notrational rational numbers. are of real numbers that are definition. set of real numbers a better definition, butintuitive it’snumbers. outside the2scope thistwo course. Formany this semester we’ll make due not this the scope thisa course. For this notion of of what real number is.) semester we’ll make due with this intuitive Numbers as subsets. Any natural number number isis an an integer. integer. Any Any integer integer isis aa Numbers as subsets. Any notion of what a real number natural is.) rational number, and any any rational number real number. number. We We can can write write rational number, and isis aa real Numbers as subsets. N çrational Z ç Q çnumber JR thismore moresuccinctly succinctly as N ⇥ Z ⇥ Q ⇥ R Numbers as subsets. this as N*✓✓ZZ* ✓✓QQ ✓R * * * * * N * ✓ *R * * * * * * * * * * ** ** ** ** ** ** 3 * * * * * * * *3* ** ** ** ** ** ** More set set notation notation More Sometimes in in math math we we find find itit more more precise precise to to use use aa common common collection collection ofof Sometimes symbols to to express express our our thoughts. thoughts. The The symbols symbols have have less less room room for for interpreinterpresymbols tation than than normal normal written written language, language, so so less less confusion confusion arises arises when when we we use use tation them. One One important important example example ofof this this isis aa method method we we use use to to describe describe sets. sets. them. Let’sstart startwith withan anexample, example,then thenwe’ll we’llexplain explainthe theexample: example: Let’s nn nn oo n,m m22ZZand andm m6=6=00 QQ== n, m m Let’s break break down down this this equation equation one one symbol symbol at at aa time. time. We We know know that that QQ isis Let’s theset setofofrational rationalnumbers, numbers,and andthe theequal equalsign signmeans meansthat thatQQisisthe thesame sameset set the asthe theset seton onthe theright rightofofthe theequal equalsign. sign. as Youknow knowthat thateverything everythingon onthe theright rightofofthe theequal equalsign signisisaaset setbecause becauseit’s it’s You all surrounded surrounded by by the the two two curly curly brackets brackets “{” “{” and and “}”, “}”, so so now now let’s let’s discuss discuss all everything in in between between the the curly curly brackets, brackets, moving moving from from left left to to right. right. The The everything n n means that the set on the right side of the equal sign is a set of fraction m fraction m means that the set on the right side of the equal sign is a set of nn means “that satisfy the fractions. The The vertical vertical line line immediately immediately following following m fractions. m means “that satisfy the 12 13 following rules”. Everything to the right of the vertical line is the collection of rules that the fractions of the set satisfy, in this case, n and m are integers, n and m 6= 0. So the set m | n, m 2 Z and m 6= 0 is the set of all fractions n m that satisfy the rules that n and m are integers and m 6= 0. Putting it all together, the expression nn o Q= n, m 2 Z and m 6= 0 m should be read as saying that the set of rational numbers is the set of fractions of integers, as long as the denominator of the fractions do not equal 0. And we already knew that. This is perhaps just a slightly more efficient way to write that. That is, it will be more efficient once you are comfortable with it, and that won’t take too long. Examples. n •Z= m | n 2 Z and m = 1 says that the integers are the same set as the set of fractions that satisfy the rules that the numerator is an integer and the denominator equals 1, and that’s true. Notice that 31 = 3 and that 3 2 Z. Similarly, 117 = 17 and 17 2 Z, etc. • E = n 2 Z | n2 2 Z says that E is the set of integers that satisfy the rule that when you divide them by 2, you still have an integer. Notice that 42 = 2 2 Z, that 218 = 9 2 Z, and that 32 2 / Z. So the above equation of sets is telling us is that E is the set of even numbers. • N = n 2 Z | n > 0 says that the natural numbers is the set of all integers that are positive. And that’s true. The positive numbers in {..., 2, 1, 0, 1, 2, 3, ...} are exactly the numbers {1, 2, 3, 4, ...}. • { x 2 {2, 7, 3} | x < 0 } = { 3} says that number from the collection of 2, 7, and 3. * * * * * * * * * * 3 is the only negative * * * Intervals Below is a list of 8 important types of subsets of R. In the sets below, a and b are real numbers. That is, a, b 2 R. Also, in the sets below a b. Any set of one of these types is called an interval. 14 a xx x bb }}b [a,{a,b] b] = = {= { xx{x 2R R |RI | aa [a, b] 2 } ( b Q a x<x (a,(a,b) b) = == 2R R |RI | aa < < x< < b< b }}b} .( (a, b) {{ xx{x 2 Oj.rn1u,imuwsvemO b 0.. [a,[a,b)={xERIa<x<b} b) = = {{ xx 2 2R R || aa xx < < bb }} ( [a, b) 411.IJIL.*1L_1*_O 0.. a x<x (a,(a,b] b] = == 2R R |RI | aa < < x bb }}b} (a, b] {{ xx{x 2 a. a xx }}x} [a,[a,oo)={x 1) = = {{ xx 2 2R R || aa [a, 1) ERI b b M$W4*W 1, b] b] = = {{ xx 2 2R R || xx bb }} (( (-oo,b]={xERIxb} 1, b (a,(a,oo)={XERIa<x} 1) = = {{ xx 2 2R R || aa < < xx }} (a, 1) (_ 0 1, b) b) = = {{ xx 2 2R R || xx < < bb }} (( (—oo,b)={XERIX<b} 1, ** * ** * ** * ** * ** * ** * t ** * ** * ** * ** * ** * ** * ** * TheEmpty The Empty Set Set EmptySet The The setset {4,{4, 7, 9} 9}9} is is the setset that contains thethe numbers 4, 4, 7, 7, and 9. 9.The The setset The the that contains 7, The set {4, 7, is the set that contains the numbers 4, 7, and 9. set numbers and The {4,{4, 7}7}contains contains only 7, 7,the the setset{4} {4} contains only 4, 4,and and thethe setset{{ }}{ } only44 and 4and and7, containsonly {4, 7} set only 4, the set the {4}contains contains only and contains, well, nothing. contains, nothing. well, contains, well, nothing. It It probably seems silly toto you now, but the setset that has nothing in in it,it, that probably seems silly now, It probably seems silly to you now, but the set that has nothing in it, that you but the that has nothing that is is the setset }, anan important example of of set, and it’sit’s set that comes upup the important is the set {{ }, important example of aa set, and it’s aa set that comes up example a set, and a set that comes { },isis isan frequently in inmath. math. It Itcomes comes upup often enough that we’ve given it it name. math.It frequently comes frequently in up often enough that we’ve given it aa name. often enough that we’ve given a name. We call it it thethe empty set, because it’sit’s thethe setset that has nothing in in it.it. call We empty set, because We call it the empty set, because it’s the set that has nothing in it. nothing that has Even more than having own name, we’ve given itsits own symbol. That’s having Even more In addition tothan having itsit’s own name, we’ve given ititits own symbol. it’s own name, we’ve given it own symbol.That’s That’s because wewewrote wrote the empty setsetas asas{{ }, }, then readers might just wonder becauseifif if wrotethe theempty emptyset because we readers then readersmight mightjust justwonder wonder { },then whether that’s typo, that perhaps wewe intended toto type something inside of of whether whether that’s aa typo, that perhaps we intended to type something inside of that’s that a typo, perhaps intended type something inside the curly brackets, but just neglected to.to.Instead, Instead, wewe useuse the symbol to the brackets,but butjust curly the curly brackets, neglected to. use the symbol ;; to just neglected Instead,we the symbol 0 to refer to the empty set. refer empty set. refer to to thethe empty set. 14 14 15 Example. If S was the set of all roots of the quadratic x2 4, and then S = { 2, 2}. If S was the set of all roots of the quadratic x2 , then S = {0}. And if S was the set of all real number roots of the quadratic polynomial 2 x + 1, then S = ;, because x2 + 1 has no real number roots. The Empty Set is a subset of any other set In the explanation a few paragraphs before, we began with the set {4, 7, 9}. We took away an object and were left with the smaller set {4, 7}. Then we took away another object and were left with the still smaller set {4}. We ended by taking away the last object to arrive at a still smaller set, namely the empty set, ;. We can think of a set B as being a subset of another set C if we can remove objects from C to obtain B. In that way we can write ; ✓ {4} ✓ {4, 7} ✓ {4, 7, 9} Of course if C is any set, it doesn’t matter which one, and if we took away all of the objects from C, then we’d be left with ;. That is to say, regardless of what the set C is. ;✓C 16 Exercises Determine whether the statements made in #1-15 are true or false. 1.) 3 2 {3, 6, 1} 2.) 1, 6 2 {3, 6, 1} 3.) 4 2 {3, 6, 1} 4.) 4 2 / {3, 6, 1} 5.) {2, 3} ✓ {3, 6, 1} 6.) {2, 3} * {3, 6, 1} 7.) ; ✓ {3, 6, 1} 8.) {3, 6, 1} 9.) [1, 3) ✓ [1, 3] 10.) 2 2 [1, 3] 11.) 1 2 (1, 3] 12.) 3 2 (1, 3] 13.) [1, 3) ✓ (1, 1) 14.) [1, 3) ✓ ( 1, 5] {6} = {3, 1} 15.) ; ✓ [1, 3) Each of the sets in #16-21 is an interval. Name the interval in each of these problems. 16.) [2, 5) (3, 5) 17.) [2, 5) (2, 5) 18.) [2, 5) [4, 5) 19.) [2, 5) {2} 20.) [2, 5) [2, 3) 21.) [2, 5) [2, 4] 17 Let T = {1, 2, 3}. Match the numbered sets on the left to the lettered sets on the right that they equal. 22.) { x 2 T | x 2 2 Z} A.) {1, 2, 3} 23.) { x 2 T | x 2 N } B.) {1, 2} 24.) { x 2 T | x > 1 } C.) {1, 3} 2 Z} D.) {2, 3} x 2 N} E.) {1} 2 / Z} F.) {2} 28.) { x 2 T | x 2 } G.) {3} 25.) { x 2 T | x 3 26.) { x 2 T | 27.) { x 2 T | 29.) { x 2 T | x 2 1 x 2 Z} H.) ; 18