Sets - Mathematics PDF

get the complete book: http://www.onemathematicalcat.org/getFullTextFullBook.htm 9. MATHEMATICIANS ARE FOND OF COLLECTIONS collections ‘Collections’ are extremely important in life: when we group together objects that are in s...

get the complete book: http://www.onemathematicalcat.org/getFullTextFullBook.htm 9. MATHEMATICIANS ARE FOND OF COLLECTIONS collections ‘Collections’ are extremely important in life: when we group together objects that are in some way similar, then it is easier to talk about the unit. The list of possible ‘collections’ goes on and on: females, democrats, students at Miss Hall’s School, your relatives, the folders in your personal filing cabinet, your favorite books,... Collections are also extremely important in mathematics. A group of similar objects can be given a name, making the group easier to refer to. Tools can be developed for working with the objects in a particular collection. Furthermore, you’ve probably noticed the fondness that mathematicians have for using let- ters (like x): every time you see such a letter, there’s a ‘collection’ associated with the letter lurking in the background. (More on this in the next section— Holding This, Holding That.) In mathematics, people study collections of numbers (like R and Z); collections of sentences; even collections of collections! The idea of ‘collection’ is made precise by the mathematical construct called a set. DEFINITION A set is a collection with the following property: given any object, either the set object is in the collection, or isn’t in the collection. understanding The key idea is this: to qualify as a set, one need only be certain that every the definition object (like the number 2 , or ‘chair’, or ‘grasshopper’) is either IN the collec- tion, or NOT IN the collection. It’s not necessary to know which of these two cases occurs (i.e., whether the object is IN or NOT IN the collection); it’s only necessary to know that exactly one of these two situations occurs! This idea is a bit subtle; a couple of examples should provide some clarification. EXAMPLE ‘The collection of some people’ is not a set. Is the author of this book in a non-set the collection? Maybe. Or maybe not. That is, given the object ‘author of this book’ (or any other person, for that matter), it is impossible to state with certainty that either the author IS in the collection, or IS NOT in the collection. Roughly, ‘vagueness’ prevents this collection from being a set. EXAMPLE Consider the collection of numbers having 3 , 6 , 9 , 12 , 15 , 18 ,... as mem- a set bers. Observe that 3 goes into each of these numbers evenly; and, there are infinitely many members in this collection. Is this a set? That is, given any object, can we definitively say that either it IS in the collection, or IS NOT in the collection? Let’s try a few: Is ‘grasshopper’ in the collection? Certainly not: it’s not even a number, so it doesn’t have a chance. Is 7 in the collection? Well, it’s a number, but 3 doesn’t go into it evenly. So, it’s not in the collection. Is 72 in the collection? It’s a number, and 3 goes into it evenly, so it is in the collection. 58 − 1 copyright Dr. Carol JVF Burns 57 http://www.onemathematicalcat.org  get the complete book: http://www.onemathematicalcat.org/getFullTextFullBook.htm Of course, it’s impossible to ‘test’ all possible objects. However, we CAN with certainty conclude that, given any object, either it’s IN the collection, or it ISN’T. For example, consider the (very large) number 35, 983, 205, 119, 780, 238, 482, 108, 222, 239, 407, 290, 981, 239. (*) Is this number in the collection? You can’t use your calculator to help you decide, because the number is way too large. And (unless you’re a very patient person) you probably don’t want to take the time to divide it by 3 , by hand. (You could, of course, use the divisibility test discussed in section 2, but even applying this test requires some patience.) However, we CAN with complete certainty say the following: either 3 goes in evenly, or it doesn’t. One or the other must happen. So, the collection is a set. mathematics One part of the previous example brings to light a subtle fact about mathemat- is primarily ics: it has primarily evolved to be a written language, not a spoken language. a written language Consequently, people sometimes run across mathematical stuff that is not con- venient to read aloud. The number in (*) is one such case. Although standard vocabulary allows us to read many large numbers aloud, (*) is so large that it goes beyond the words supplied. If forced to read (*) aloud, then most people would either change its name to one more suited to large numbers (scientific notation), or else say: ‘the large number whose digits are: three, five, comma; nine, eight, three, comma;... ’. a set is an Whenever you are presented with any new mathematical concept, you should EXPRESSION, immediately address its most primary classification: is it an expression, or a not a sentence sentence? A set is a mathematical expression: it is a name given to some collection of interest. It doesn’t make sense to ask if a set is TRUE or FALSE, because a set is not a sentence. Sets (like expressions in general) can have lots of different names. The remainder of this section is devoted to notation used in connection with sets. members of a set; The objects in a set are called its elements, or its members (the two terms elements of a set are used interchangeably). A set can have no members (0 members), 1 member, 2 members, 3 members, etc. If a set has n members, where n is a whole number, finite set; then it is called a finite (FI–nite) set. Otherwise, it is called an infinite (IN– infinite set fi–nit) set. For example, a set with 203 members is a finite set. A set with no members is a finite set. The set of integers that lie between −3 and 985 is a finite set. The set of all integers is an infinite set. The set of real numbers is an infinite set. The set of all real numbers between 2 and 3 is an infinite set. EXERCISES 1. Sometimes a definition is embedded in a paragraph, instead of putting it in a nice box labeled DEFINITION. This just happened. You were actually given four definitions: element (of a set), member (of a set), finite set, infinite set. State the definitions of these four things. 2. Consider the sentence: ‘n is a whole number.’ This sentence can be true or false, depending upon the number chosen for n. What value(s) of n make the sentence true? False? list method Some sets (but not all) can be easily described using the list method: in this for naming method, the members are separated by commas, and enclosed in braces { }. (some) sets 57 + 1 copyright Dr. Carol JVF Burns 58 http://www.onemathematicalcat.org  get the complete book: http://www.onemathematicalcat.org/getFullTextFullBook.htm the order of the For example, the set {0, 1, 2} has three members: 0 is a member, 1 is a member, members in the list and 2 is a member. When using the list method with a finite number of ele- doesn’t make ments, the order in which the elements are listed doesn’t make any difference. a difference Therefore, {0, 1, 2} and {0, 2, 1} and {1, 0, 2} and {1, 2, 0} and {2, 0, 1} and {2, 1, 0} are all just different names for the same set. Notice that there are 3 · 2 · 1 = 6 ways to rearrange the three elements in this set. How is ‘ {0, 1, 2} ’ You can read ‘ {0, 1, 2} ’ as: read aloud? ‘the set with members 0 , 1 , and 2 ’ ; or ‘open brace, zero, one, two, close brace’. The first way expresses an understanding of the symbols; the second is a ‘literal’ reading of each symbol. EXERCISES 3. It’s important to be able to write braces { } correctly. (Braces are ‘curly’, like the things on teeth.) In particular, braces must be easy to distinguish from parentheses ( ) and brackets [ ]. Trace the following as practice: { } { } { } { } { } { } ( ) ( ) ( ) ( ) ( ) ( ) [ ] [ ] [ ] [ ] [ ] [ ] 4. What do the centered dots in the expression 3 · 2 · 1 mean? 5. How many members are in the set {a, b, c, d, e} ? Make a conjecture (i.e., an educated guess) as to how many different re-arrangements of this set there are. A set can have The set {... , −3, −2, −1, 0, 1, 2, 3,... } has an infinite number of members. lots of different names! Indeed, this is the set of integers, denoted by the symbol Z. Since Z and {... , 3, −2, −1, 0, 1, 2, 3,... } are just different names for the same set, the sentence Z = {... , −3, −2, −1, 0, 1, 2, 3,... } is true. there are sets Not all sets can be described using the list method. For example, the real that can’t be described numbers shaded below can’t be described using the list method. You could using the list method certainly try: {2 , 2.001 , 2.002 , 2.003 ,... , 3} ; but you’d be missing infinitely many numbers between, say, 2.001 and 2.002. 2 3 a verb to discuss To talk about membership in a set, we need the verb ∈ , which is called the membership in a set: ‘ is in ’ or ‘ is an element of ’ or ‘ is a member of ’ symbol. A precise discussion ∈ of this verb follows. 2 · 25 + 3 · 3 copyright Dr. Carol JVF Burns 59 http://www.onemathematicalcat.org  get the complete book: http://www.onemathematicalcat.org/getFullTextFullBook.htm sentence: Let x represent any object, and let S represent any set. The sentence x∈S x∈S is read as: ‘ ex is in S ’ or ‘ ex is an element of S ’ or ‘ ex is a member of S ’. These three phrases are used interchangeably. naming conventions The sentence ‘ x ∈ S ’ illustrates a couple naming conventions for sets. Firstly, for sets: the letter S is commonly used to name sets, since it is the first letter in the word sets are named with ‘set’. Secondly, sets are usually represented by uppercase (capital) letters— capital letters; more on this in the section Holding This, Holding That. Furthermore, this is a good time to mention that, in mathematics, uppercase and lowercase letters uppercase and lowercase are NOT interchangeable: the lowercase (like t) and uppercase (like T ) versions are NOT interchangeable of letters usually represent totally different objects. When is the sentence When is the sentence ‘ x ∈ S ’ true? False? ‘ x ∈ S ’ true? If x really IS a member of S , then the sentence ‘ x ∈ S ’ is TRUE. And, if the False? sentence ‘ x ∈ S ’ is TRUE, then x must be a member of S. Similarly, the sentence ‘ x ∈ S ’ is FALSE precisely when x IS NOT a member of S. Before looking at some examples, it’s necessary to point out a sentence structure that is commonly used in mathematics when something is being given a name. What does the phrase In mathematics, the phrase ‘ LET S = {1, 2, 3}’ LET S = {1, 2, 3} mean? means: take the set {1, 2, 3} and give it the name S, so that it will be easier to refer to. More generally, a sentence of the form LET NAME = EXPRESSION is used to give the name NAME to the expression EXPRESSION. The word ‘LET’ is the key to knowing that something is being named. Here are some examples: Let x = 4.217. (The name x is being given to the number 4.217.) Let W = {0, 1, 2, 3,... }. (The name W is being given to the set of whole numbers.) Let t = 21 + 31. (The name t is being given to the sum of 12 and 13.) EXAMPLE Here are some examples using the verb ‘ ∈ ’. Let S = {1, 2, 3}. Then, the using the verb ‘ ∈ ’ following sentences are all true: 6 16−7 1∈S 2∈S 3∈S 3 ∈S 3 ∈S 6 16−7 (Note: is just another name for 2 ; and 3 3 is just another name for 3.) The following sentences are false: 4∈S 1.000001 ∈ S 0∈S EXERCISES 6. In mathematics, how would you say: ‘Take the set {a, b, c, d, e}, and give it the name T ’ ? 7. In mathematics, how might you more compactly say: ‘Take all the whole numbers greater than or equal to 7, and give this collection the name S ’ ? 6(1 + 2 + 3 + 4) copyright Dr. Carol JVF Burns 60 http://www.onemathematicalcat.org  get the complete book: http://www.onemathematicalcat.org/getFullTextFullBook.htm EXERCISES 8. Let W = {3, 4, 5,... }. Decide whether the following sentences are true, false, or sometimes true/sometimes false (ST/SF): (a) 3 ∈ W (b) 107 ∈ W (c) 83 ∈ W (d) 93 ∈ W (e) x ∈ W Step 1: Step 2:................................................................................................................................................. 9. It’s important to be able to write the symbol ∈ correctly. Trace the................................................................. following as practice: ∈ ∈ ∈ ∈ ∈ ∈ ∈ ∈ ∈ ∈ ∈ 10. Classify each entry below as an expression or a sentence. If an expression, state whether it’s a number or a set. If a sentence, state how you might read it aloud, and state whether it is true, false, or ST/SF. (a) 5 (b) {5} (c) Z (d) 5 ∈ R (e) 5.1 ∈ Z the ‘not in’ verb: ∈ / A verb is frequently negated by putting a slash through it. Consequently, the the sentence: x ∈ /S sentence ‘ x 6∈ S ’ is read as ‘ ex is not in S ’ or ‘ ex is not an element of S ’ or ‘ ex is not a member of S ’. The sentence ‘ x ∈ / S ’ is true when x is not a member of S; it is false otherwise. For example, the sentence ‘ 5 ∈ / {1, 2, 3} ’ is true. a slash / is used Here are more examples of using a slash to negate a verb: to negate action verb how to read negated verb how to read ∈ is in ∈ / is not in ∈ is an element of ∈ / is not an element of ∈ is a member of ∈ / is not a member of = is equal to 6= is not equal to = equals 6= does not equal > is greater than 6> is not greater than < is less than 6< is not less than Notice the variety of ways that some symbols can be read. EXERCISE 11. For each sentence below, make a number line, and shade the value(s) of x that make the sentence true. Be careful to distinguish between hollow dots (numbers not included) and solid dots (numbers included). (a) x ∈ {0, 3, 5} (b) x is a real number and x 6= 2 (c) x is a real number and x ∈/ {0, 3, 5} 60.9 copyright Dr. Carol JVF Burns 61 http://www.onemathematicalcat.org  get the complete book: http://www.onemathematicalcat.org/getFullTextFullBook.htm the empty set There is exactly one set that has NO members: it is appropriately called the has no members: empty set, and is denoted using either the symbol Ø , or a pair of braces with Ø nothing inside: { }. Consequently, the sentence x ∈ Ø (or x ∈ { } ) is always {} FALSE, since the empty set has no members! The astute reader may have noticed the similarity between the symbols ∅ (some- times used for the number zero) and Ø (the empty set). Context will help to clarify the correct interpretation, since numbers and sets get used in different types of places. EXERCISE 12. State how you might read each sentence. Also, classify each sentence as true, false, or ST/SF: (a) 1 ∈ { } (b) 0 ∈ Ø (c) 0 ∈/{} (d) x ∈/Ø intervals As mentioned earlier, not all sets can be listed. Indeed, there is an important class of frequently-used sets, called intervals, that cannot be listed. The def- inition of an interval is given next, and then an important notation used to describe intervals. DEFINITION An interval is a set of real numbers that has one of the following forms: interval.......................................... two endpoints, neither endpoint included..................... two endpoints, only left-hand endpoint included..................... two endpoints, only right-hand endpoint included two endpoints, both endpoints included.................................................................... one endpoint, not included, with everything to its right................................................ one endpoint, included, with everything to its right........................................................................................... one endpoint, not included, with everything to its left.......................................................................................................................................... one endpoint, included, with everything to its left......................................................................................................................................................................................... Everything! F The empty set and singletons are sometimes considered to be intervals. The empty set is a ‘degenerate’ form of an open interval (a, b) , when a = b. A singleton is a ‘degenerate’ form of a closed interval [a, b] , when a = b. interval notation: Intervals are very common in mathematics, so there is a special notation for naming them, which is appropriately called interval notation. Notice carefully the difference between the use of parentheses ( ) and brackets [ ] in the following ∞ ‘infinity’ examples. The symbol ∞ is read as ‘infinity’ (in–FIN–i–tee), and the symbol −∞ ‘negative infinity’ −∞ is read as ‘negative infinity’. start here ↓................................................................................................................................................................................................. Drawing the ‘infinity’ symbol............................................................................................................................................................................................................................ P∞ 9 61 + j=1 10j copyright Dr. Carol JVF Burns 62 http://www.onemathematicalcat.org  get the complete book: http://www.onemathematicalcat.org/getFullTextFullBook.htm examples SET NAME USING of interval notation INTERVAL NOTATION:........................................ (1, 2) −2 −1 0 1 2..................... [1, 2) −2 −1 0 1 2 [1, 2] −2 −1 0 1 2..................................................................... (1, ∞) −2 −1 0 1 2................................................ [1, ∞) −2 −1 0 1 2..................................................................... (−∞, 2) −2 −1 0 1 2................................................ (−∞, 2] −2 −1 0 1 2................................................................................................................................................................................................................................................................................... (−∞, ∞) −2 −1 0 1 2 rules for Here are the rules for interval notation: interval notation WHEN THERE ARE TWO ENDPOINTS: List the endpoints of the interval, separated by commas; always list the endpoints in order from left to right on the number line. If an endpoint is to be INCLUDED, put a bracket [ ] next to it. If an endpoint is NOT to be included, put a parenthesis ( ) next to it. WHEN THERE IS ONLY ONE ENDPOINT: The symbol ∞ is used when you are to continue forever to the right. The symbol −∞ is used when you are to continue forever to the left. ∞ is NOT Note that ∞ is NOT a real number: that is, there is no point on the number a real number! line corresponding to ∞. Instead, ∞ suggests the idea that the number line extends infinitely far to the right: given any real number (no matter how far to the right of zero it lives), there is always a real number that lives farther to the right! (What does −∞ suggest?) Consequently, parentheses ( ) are always used with the symbols ∞ or −∞, since they can’t be included. MEMORY DEVICE Brackets [ ] have sharp corners—and dust collects in corners—so brackets cor- respond to FILLED-IN endpoints. (Imagine the endpoint filled with dust!) On the other hand, parentheses ( ) do NOT have corners, so dust can’t collect here; parentheses correspond to HOLLOW endpoints. reading interval notation Unfortunately, there’s not always a great way to read interval notation aloud— another illustration that mathematics is primarily a written language, not a spoken language. Here are some possibilities at this stage in your mathematical career: ∞ X 1
62 + 9 ( )k copyright Dr. Carol JVF Burns 63 10 k=1 http://www.onemathematicalcat.org  get the complete book: http://www.onemathematicalcat.org/getFullTextFullBook.htm line # INTERVAL POSSIBLE WAY TO READ ALOUD 1 (2, 3) the real numbers between 2 and 3 , not including the endpoints 2 (2, 3) parenthesis... 2... comma... 3... parenthesis 3 [2, 3) the real numbers between 2 and 3 , including 2 but not 3 4 [2, 3) bracket... 2... comma... 3... parenthesis 5 (2, ∞) the real numbers greater than 2 6 [2, ∞) the real numbers greater than or equal to 2 7 (−∞, 2) the real numbers less than 2 8 (−∞, 2] the real numbers less than or equal to 2 Lines 1 and 3 display an understanding of the symbols; lines 2 and 4 are ‘literal’ or ‘verbatim’ readings of the symbols. Lines 5 through 8 use the words ‘greater than’ and ‘less than’, which are thoroughly discussed in the section I Live Two Blocks West Of You. EXERCISE 13. Classify each entry below as an expression or a sentence. If an expression, state whether it is a number or a set. If a sentence, state how you might read it aloud, and state whether it is true, false, or ST/SF. (a) {1, 2} (b) [1, 2] (c) 1 + 2 (d) (1, 2] (e) 1 ∈ (1, 2] (f) 1 ∈ [1, 2) set-builder There are sets that cannot easily be described using either the list method, or notation interval notation. In such cases, a naming scheme called ‘set-builder notation’ usually comes to the rescue. (Set-builder notation will not be discussed in this book.) subset Sometimes, it is necessary to discuss various subcollections chosen from a given set. This idea of subcollection is made precise as follows: DEFINITION Let S be a set. Set B is called a subset of S if any one of the following three subset conditions holds: (a) B is the set S itself (b) B is the empty set (c) each member of B is also a member of S understanding The first sentence, the definition ‘Let S be a set.’ means ‘Let S be any set.’ So why don’t mathematicians say ‘any’, if they mean ‘any’ ? Well, the word ‘a’ is shorter than the word ‘any’—and mathematicians are an extremely frugal lot. (1 + 1)6 copyright Dr. Carol JVF Burns 64 http://www.onemathematicalcat.org  get the complete book: http://www.onemathematicalcat.org/getFullTextFullBook.htm investigating Now, let’s investigate conditions (a)—(c). Remember that these are the condi- (a)–(c) tions under which B gets to be called a subset of S. Condition (a) tells us that a set is a subset of itself; this is the ‘subcollection’ consisting of everything in the original set. Condition (b) tells us that the empty set is a subset of every set; this is the ‘subcollection’ consisting of nothing from the original set. By combining conditions (a) and (b), we see that every set (except the empty set) is guaranteed to have at least two subsets: itself and the empty set. The next example explores condition (c), and shows that most sets have lots of subsets: EXAMPLE Let S = {1, 2, 3}. All the subsets of S are listed below: {1, 2, 3} (S itself) {} (the empty set) {1} {2} {3} (all one-member subsets) {1, 2} {1, 3} {2, 3} (all two-member subsets) Thus, the set {1, 2, 3} (or any set with three members) has eight subsets. EXERCISES 14. List all the subsets of {a, b}. How many subsets are there? 15. List all the subsets of {0, 2, 4}. How many subsets are there? 16. Justify your answers to each of the following questions: (a) Is {−1, 2, 3} a subset of R ? (b) Is {−1, 2, 3} a subset of the whole numbers? (c) Is {−1, 2, 3} a subset of the integers? (d) Is {−1, 2, 3} a subset of (−2, ∞) ? 5 · 13 copyright Dr. Carol JVF Burns 65 http://www.onemathematicalcat.org  get the complete book: http://www.onemathematicalcat.org/getFullTextFullBook.htm END-OF-SECTION For exercises 17–23: Classify each entry as a mathematical expression (EXP), EXERCISES or a mathematical sentence (SEN). If an expression, state whether it is a number or a set. Classify the truth value of each sentence: (always) true (T); (always) false (F); or sometimes true/sometimes false (ST/SF). 1 1 1 17. {1, , , ,... } 2 3 4 1 1 1 1 18. ∈ {1, , , ,... } 100 2 3 4 1 1 1 19. 0.01 ∈ {1, , , ,... } 2 3 4 20. (3, 5] 21. 3 ∈ (3, 5] 22. 5 ∈ (3, 5] 23. 4.997 ∈ (3, 5] Describe the following sets of numbers using correct set notation. Use either list or interval notation; whichever is appropriate. 24. −2 −1 0 1 2 25..................... −2 −1 0 1 2 26...................... −2 −1 0 1 2 27................................................. −2 −1 0 1 2 28...................................................................... −2 −1 0 1 2 29. List all subsets of the set in exercise (24). 30. Is the set of positive integers a subset of the set in exercise (28)? Justify your answer. EXERCISES Go to my homepage http://onemathematicalcat.org and navigate to my web practice Algebra I course, which has about 170 sequenced lessons. It can be used as a complete year-long high school course, or one semester in college. You’re currently looking at the pdf version—you’ll see that the HTML version has unlimited, randomly-generated, online and offline practice in every section. It’s all totally free. Enjoy! 1 · 11 + 2 · 11 + 3 · 11 copyright Dr. Carol JVF Burns 66 http://www.onemathematicalcat.org  get the complete book: http://www.onemathematicalcat.org/getFullTextFullBook.htm SECTION SUMMARY MATHEMATICIANS ARE FOND OF COLLECTIONS NEW IN THIS SECTION HOW TO READ MEANING set A collection satisfying: given any object, either the object is is the collection, or isn’t in the collection. A set is an expres- sion. members; elements the objects in a set finite set FI–nite a set with n members, where n is a whole number infinite set IN–fi–nit a set that is not finite list method a method for naming sets whose elements can be listed {, } open brace; close brace; used in list notation; the elements are listed, plural is ‘braces’ separated by commas, inside the braces ∈ ‘ is in ’ verb used to talk about membership in a ‘ is an element of ’ set ‘ is a member of ’ x∈S ‘ x is in S ’ sentence: true when x is a member of the ‘ x is an element of S ’ set S ; false otherwise ‘ x is a member of S ’ Let NAME = EXPRESSION used whenever you want to assign the name NAME to the expression EXPRESSION / (forward) slash used to negate a verb x∈ /S ‘ x is not in S ’ sentence: true when x is not a member of S ; false otherwise Ø or { } the empty set the unique set that has no members (a, b).......................................... the real numbers be- intervals of real numbers a b tween a and b (with (a, b].............. a....... various endpoints in- cluded/not included) b [a, b) a..................... b [a, b] a b 100 − 33 copyright Dr. Carol JVF Burns 67 http://www.onemathematicalcat.org  get the complete book: http://www.onemathematicalcat.org/getFullTextFullBook.htm NEW IN THIS SECTION HOW TO READ MEANING (a, ∞).................................................................... the real numbers greater intervals of real numbers a than a ; greater than [a, ∞) a............................................... or equal to a (−∞, b)........................................................................................... the real numbers less intervals of real numbers b than b ; less than or (−∞, b]...................................................................... equal to b b (, ) open parenthesis; when used in interval notation, denotes close parenthesis; that an endpoint is NOT to be included; plural is ‘parentheses’ a parenthesis is always used next to ∞ or −∞ [, ] open bracket; when used in interval notation, denotes close bracket; that an endpoint IS to be included plural is ‘brackets’ ∞ , −∞ infinity; negative infinity The symbol ∞ suggests the idea that given any real number, no matter how far to the right of zero, there is always one farther to the right. The symbol −∞ suggests the idea that given any real number, no matter how far to the left of zero, there is always one far- ther to the left. subset Set B is a subset of a set S if one of the following conditions holds: B = S, B = Ø, or each member of B is also a member of S. 22 · 17 copyright Dr. Carol JVF Burns 68 http://www.onemathematicalcat.org  get the complete book: http://www.onemathematicalcat.org/getFullTextFullBook.htm SOLUTIONS TO EXERCISES: MATHEMATICIANS ARE FOND OF COLLECTIONS IN-SECTION EXERCISES: 1. The object(s) in a set are called the element(s) or member(s) of the set. If a set has n members, where n is a whole number, then it is a finite set. Otherwise, it is an infinite set. 2. The sentence ‘ n is a whole number ’ is true when n is chosen from the set {0, 1, 2, 3,... }. Otherwise, the sentence is false. 4. The centered dots denote multiplication. 5. There are 5 members in the set {a, b, c, d, e}. There are 5 · 4 · 3 · 2 · 1 = 120 different re-arrangements of the five-member set {a, b, c, d, e}. 6. Let T = {a, b, c, d, e}. 7. Let S = {7, 8, 9,... }. 8a. T 8b. T 8c. F 8d. T 8e. ST/SF 10a. EXP, number 10b. EXP, set 10c. EXP, set 10d. SEN, T Read as: ‘ 5 is a member of the set of real numbers’ or ‘ 5 is in the set of real numbers’ or ‘ 5 is an element of the set of real numbers’ or, most simply, ‘ 5 is a real number’ 10e. SEN, F Read as: ‘ 5.1 is a member of the set of integers’ or, most simply, ‘ 5.1 is an integer’ 11a. 0 3 5 11b.................................................................................................................................................................. 2 11c............................................................................................................................................................................. 0 3 5 12a. ‘ 1 is in the empty set ’; false 12b. ‘ 0 is in the empty set ’; false 12c. ‘ 0 is not in the empty set ’; true 12d. ‘ x is not in the empty set ’; (always) true 13a. EXP, set 13b. EXP, set 13c. EXP, number 13d. EXP, set 13e. SEN, F ‘ 1 is in the the set of real numbers between 1 and 2 , including 2 but not including 1 ’ 13f. SEN, T ‘ 1 is in the set of real numbers between 1 and 2 , including 1 but not including 2 ’ 14. The subsets of {a, b} are: {a, b} , Ø , {a} , {b}. There are four subsets. 15. The subsets of {0, 2, 4} are: {0, 2, 4} , Ø , {0} , {2} , {4} , {0, 2} , {0, 4} , {2, 4}. There are eight subsets. 3 · 23 copyright Dr. Carol JVF Burns 69 http://www.onemathematicalcat.org  get the complete book: http://www.onemathematicalcat.org/getFullTextFullBook.htm 16a. The set {−1, 2, 3} is a subset of R , since every member of {−1, 2, 3} is a real number. 16b. The set {−1, 2, 3} is not a subset of the whole numbers, because −1 is not a whole number. 16c. The set {−1, 2, 3} is a subset of the integers, since every member of {−1, 2, 3} is an integer. 16d. The set {−1, 2, 3} is a subset of the interval (−2, ∞), since every member of {−1, 2, 3} is in the interval (−2, ∞). END-OF-SECTION EXERCISES: 17. EXP, set 18. SEN, T 19. SEN, T 20. EXP, set 21. SEN, F 22. SEN, T 23. SEN, T 24. {−1, 0, 1} 25. (−1, 1] 26. [0, 2) 27. (−∞, 1] 28. (−1, ∞) 29. There are eight subsets of {−1, 0, 1} : {−1, 0, 1} , Ø , {−1} , {0} , {1} , {−1, 0} , {−1, 1} , {0, 1}. 30. The set of positive integers is a subset of (−1, ∞) , since every positive integer is in the interval (−1, ∞). 4 X 7 i copyright Dr. Carol JVF Burns 70 i=1 http://www.onemathematicalcat.org