20C Sets and Venn diagrams

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1 20C Sets and Venn diagrams set is just a collection of objects, but we need some new words and symbols and diagrams to be able to talk sensibly about sets. Some tricky counting problems can be solved by using a type of diagram called a Venn diagram to illustrate the sets involved. Describing and naming sets In day-to-day life, we try to make sense of the world we live in by classifying collections of things. English has many words for such collections. For example, we speak of a flock of birds, a herd of cattle, a swarm of bees and a colony of ants. We do a similar thing in mathematics, and classify numbers, geometrical figures and other things into collections that we call sets. The objects in these sets are called the elements of the set. Describing a set set can be described by listing all of its elements. For example: S = {, 3, 5, 7, 9} We read this as S is the set with elements, 3, 5, 7 and 9. Notice how the five elements of the set are separated by commas and the list is enclosed between curly brackets. set can also be described by writing a description of its elements between curly brackets. Thus the set S can also be written as S = {odd whole numbers less than 0}, which we read as S is the set of odd whole numbers less than 0. set must be well defined It is important that our description of the elements of a set is clear and unambiguous. For example, {tall people} is not a set, because people tend to disagree about what tall means. set must be well defined, like the following set: {letters in the English alphabet}. Material from ICE-EM Mathematics, 3, 4. rown, Evans et al (2006/7)

2 Equal sets Two sets are called equal if they have exactly the same elements. Thus {a, e, i, o, u} = {vowels in the English alphabet}. On the other hand, the sets {, 3, 5} and{, 2, 3} are not equal, because they have different elements. This is written as {, 3, 5} {, 2, 3}. The order in which the elements are written between the curly brackets does not matter at all. For example: {, 3, 5, 7, 9} = {3, 9, 7, 5, } = {5, 9,, 3, 7} If an element is listed more than once, it is only counted once. For example, {a, a, b} = {a, b} The set {a, a, b} has only the two elements a and b. The second mention of a is an unnecessary repetition and can be ignored. However, it is normally considered poor notation to list an element more than once. The symbols and The phrase is an element of occurs so often in discussing sets that the special symbol is used for it. For example, if = {3, 4, 5, 6}, then 3. (We read 3 as 3 is an element of the set.) The symbol means is not an element of. For example, if = {3, 4, 5, 6}, then 8. (We read 8 as 8 is not an element of the set.) Describing and naming sets set is a collection of objects, called the elements of the set. set must be well defined, meaning that its elements can be described or listed without ambiguity. For example: {, 3, 5} and {letters of the English alphabet}. Two sets are called equal if they have exactly the same elements. If a is an element of a set S, we write a S. If b is not an element of a set S, we write b S. Material from ICE-EM Mathematics, 3, 4. rown, Evans et al (2006/7) 2

3 Exercise List the elements of each of these sets. a {odd whole numbers between 0 and 2} b {two digit numbers ending with 3} c the set of months ending in y d {continents} e {multiples of 6 between 0 and 40} f {multiples of 7 between 0 and 40} g {perfect squares less than 50} h {prime numbers less than 30} 2 Give a verbal description of each set. a {a, b, c} b {, 3, 5} c {2, 3, 5, 7} 3 State whether or not the following sets are well defined, and give a reason in each case if not. a {best films of 955} b {States of ustralia} c {whole numbers less than } d {brightly coloured birds} f {people now in this room} e {small numbers} g {people who vote Labor} 4 Which of the following pairs of sets are equal? {2, 3, 5} and {3, 5, 2} {7,, 4} and {, 9, 4} C {64, 7 + 2, } and {2, 9, } D {letters of the alphabet following v} and {w, x, w, y, z, x, w} 5 Replace in each statement by either = or. a {a, b, c} {c, b, a} b {, 2, 4} {, 4, 2,, 4, 4} c {a, b, c} {x, y, z} d {7 + 2, 5 } {4, 9} e {citizens of ustralia} {men, women and children in ustralia} 3 Material from ICE-EM Mathematics, 3, 4. rown, Evans et al (2006/7)

4 6 Let S = {8, 9, 0,, 2}. Rewrite each sentence in symbols, using or. a 9 is an element of S. b 2 is an element of S. c 5 is not an element of S. d 0 is not an element of S. 7 Replace in each statement by either or. a 3 {0,, 2, 3, 8} b dog {five-letter words} c 5 {0,, 2, 4} d Sydney {cities north of the equator} e g {vowels} f 20 {perfect squares} g {prime numbers} h 6 {even whole numbers} Finite and infinite sets ll the sets we have seen so far have been finite sets, meaning that we can list all their elements. Here are two more examples: {whole numbers between 2000 and 2005} = {200, 2002, 2003, 2004} {whole numbers between 2000 and 3000} = {200, 2002, 2003,, 2999} The three dots in the second example stand for the other 995 numbers in the set. We could have listed them all, but to save space we used dots instead. This notation can only be used if it is clear what it means, as in this case. Infinite sets set can also be infinite all that matters is that it is well defined. Here are two examples of infinite sets: {even whole numbers} = {0, 2, 4, 6, 8, 0, } {whole numbers greater than 2000} = {200, 2002, 2003, 2004, } oth these sets are infinite because no matter how many elements we list, there are always more elements in the set that are not in our list. This time the dots have a slightly different meaning, because they stand for infinitely many elements that we could not possibly list, no matter how long we tried. Material from ICE-EM Mathematics, 3, 4. rown, Evans et al (2006/7) 4

5 The symbol S for the number of elements of a finite set If S is a finite set, the symbol S stands for the number of elements of S. For example: If S = {, 3, 5, 7, 9}, then S = 5. If = {00, 002, 003,, 3000}, then = If T = {letters in the English alphabet}, then T = 26. One-element sets set may have only one element, like the set S defined by S = {5}, S is the set whose only element is 5. It is important to distinguish between the number 5 and the set S = {5}: 5 S but 5 S. In this case S =. tin containing a single biscuit is quite a different object from the biscuit itself. The biscuit tin can be thought of as a set with one element, whereas the biscuit is a biscuit. The empty set The symbol represents the empty set, which is the set that has no elements at all. The empty set is like a biscuit tin after all the biscuits have been eaten: there is nothing in the whole universe that is an element of : = 0 and x, no matter what x may be. There is only one empty set, because any two empty sets have exactly the same elements (that is, no elements) and so they must be equal to one another. Finite and infinite sets set is called finite if we can list all of its elements. n infinite set has the property that no matter how many elements we list, there are always more elements in the set that are not in our list. If S is a finite set, the symbol S stands for the number of elements of S. The set with no elements is called the empty set, and is written as. Thus = 0. 5 Material from ICE-EM Mathematics, 3, 4. rown, Evans et al (2006/7)

6 Exercise Is each of these a well defined set? If it is well defined, is it finite or infinite? a {large whole numbers} b {adults in ustralia who like music} c {fractions between and 2} d {working mobile phones in ustralia} e {points on a particular line } f {past and present ustralian senators} g {whole numbers between million and trillion} 2 Find the size of each set. a = {5, 6, 8} b = {0,, 4, 5, 8, 9} c = {0, 20, 30,, 200} d = {letters of the alphabet} e = {states of ustralia} f = {primes less than 20} g = {perfect squares less than 20} h = {50, 5, 52,, 70} i = k = {0,, } j = {p} l = {vowels} 3 List, or begin to list, the elements of each of the following sets (you may need to use dots). Then state whether it is a finite or an infinite set. a {whole numbers less than 0} b {odd whole numbers between 0 and 20} c {all perfect squares} d {whole numbers greater than 50} e {even whole numbers between and 7} f {multiples of 7 greater than 50} g {prime numbers less than 30} Material from ICE-EM Mathematics, 3, 4. rown, Evans et al (2006/7) 6

7 4 State whether each set is the empty set or is a non-empty set. a {dogs 50 m high} b {whole numbers greater than 4 but less than 6} c {English speakers on Mars} d {whole numbers greater than 7 but less than 8} e {English words with three Es} f {letters in the alphabet that are not vowels or consonants} 5 Explain whether each statement is true or false. a The set of points on a line is a finite set. b {whole numbers less than } is a finite set. c {human beings who have ever lived} is an infinite set. d If two finite sets are equal, then they have the same number of elements. e If two finite sets have the same number of elements, then they are equal. f If and are empty sets, then =. 6 Should the symbol in each statement be replaced by = or by? a If = {7, 0, 38} and = {Peter, Paul, Mary}, then. b If = {7, 0, 38} and = {Peter, Paul, Mary}, then. c {, 3, 5, 7} {7, 5,, 3} d {8, 3, 5, 7} {7, 5, 7, 8, 3, 8, 5} e If T = {primes less than 0}, then T 5. f If H = {positive multiples of 3 less than 20}, then H 6. g If K = {0,, 2,, 0}, then K 0. h If L = {0}, then L 0. i If = {0, 0}, then. j If =, then 0. 7 Material from ICE-EM Mathematics, 3, 4. rown, Evans et al (2006/7)

8 C Subsets of a set Sets of things are often further subdivided. For example, owls are a particular type of bird, so every owl is also a bird. We express this by saying that the set of owls is a subset of the set of birds. set S is called a subset of another set T if all the elements of S are elements of T. This is written as S T. (We read this as S is a subset of T.) The new symbol means is a subset of. Thus {owls } {birds} because every owl is a bird. Similarly, if = {2, 4, 6} and = {0,, 2, 3, 4, 5, 6}, then, because every element of is an element of. The sentence S is not a subset of T is written as S T. This means that at least one element of S is not an element of T. For example, {birds} {flying creatures} because an ostrich is a bird, but it does not fly. Similarly, if = {0,, 2, 3, 4} and = {2, 3, 4, 5, 6}, then, because 0, but 0. The set itself and the empty set are always subsets ny set S is always a subset of itself, because every element of S is an element of S. For example: {birds} {birds} If = {, 2, 3, 4, 5, 6}, then. Furthermore, the empty set is a subset of every set S, because every element of the empty set is an element of S, as there is no element in that lies outside S. For example: {birds} If = {, 2, 3, 4, 5, 6}, then. Subsets and the adjective all Statements about subsets can be rewritten as sentences using the word all. For example: {owls} {birds} means ll owls are birds. {birds} {flying creatures} means Not all birds are flying creatures. Material from ICE-EM Mathematics, 3, 4. rown, Evans et al (2006/7) 8

9 Venn diagrams Diagrams make mathematics easier because they help us to see the whole situation at a glance. The English mathematician John Venn ( ) began using diagrams to represent sets. His diagrams are now called Venn diagrams. The universal set In most problems involving sets, it is convenient to choose a larger set that contains all of the elements in all of the sets being considered. This larger set is called the universal set, and is usually given the symbol E. In a Venn diagram, the universal set is generally drawn as a large rectangle, and then other sets are represented by circles within this rectangle. For example, If V = {vowels}, we could choose the universal set to be E = {letters of the alphabet} and all the letters of the alphabet would then need to be placed somewhere within the rectangle, as shown below. E V a e i o u b c d f g h j k l m n p q r s t v w x y z In the Venn diagram below, the universal set is E = {0,, 2, 3, 4, 5, 6, 7, 8, 9, 0}, and each of these numbers has been placed somewhere within the rectangle. E The region inside the circle represents the set of odd whole numbers between 0 and 0. Thus we place the numbers, 3, 5, 7 and 9 inside the circle, because = {, 3, 5, 7, 9}. Outside the circle we place the other numbers 0, 2, 4, 6, 8 and 0 that are in E but not in. 9 Material from ICE-EM Mathematics, 3, 4. rown, Evans et al (2006/7)

10 Representing subsets on a Venn diagram When we know that S is a subset of T, we place the circle representing S inside the circle representing T. For example, let S = {0,, 2}, and T = {0,, 2, 3, 4}. Then S is a subset of T, as illustrated in the Venn diagram below. E 4 S T Subsets of a set If all the elements of a set S are elements of another set T, then S is called a subset of T. This is written as S T. If at least one element of S is not an element of T, then S is not a subset of T. This is written as S T. For any set S, S and S S. Statements about subsets can be rewritten using the words all or not all. Subsets can be represented using a Venn diagram. Exercise C Choose whether or should replace in these sentences. a {cats} {mammals} b {animals} {mammals} c {primes} {odd numbers} d {brick buildings} {homes} e {boys} {males} f {adults} {people} 2 Let = {, 2, 3, 4,, 0}, = {, 2, 3, 8, 9} and C = {2, 3, 5, 7, 9}. Copy these statements, replacing with either or. a b C c C d {, 4, 9, 6} e {3, 5} f g h i j C {prime numbers} k {2, 4, 6} Material from ICE-EM Mathematics, 3, 4. rown, Evans et al (2006/7) 0

11 3 Explain why each statement below is true. a If S is a set, then S. b If and C, then C. c If S is a set, then S S. d If and, then =. e If x S, then {x} S. f If x S and y S, then {x, y} S. g If, where and are finite sets, then. 4 re the statements below true or false? Explain your answers. a If and and are finite sets, then <. b If and and are finite sets, then <. c If = and and are finite sets, then =. d If C and, and C are finite sets, then C. 5 a List all subsets of each set listed below. (Recall that the set itself and the empty set are subsets.) i {a, b, c} ii {a, b} iii {a} b How many subsets did each set have? What pattern do those numbers form? 6 Rewrite each of the following statements using set notation and the symbol or. a ll fish are animals. b Numbers ending in 6 are all even. c Not all children with blonde hair are naughty. d Not all numbers ending in 4 are multiples of 4. e Not all adults are sensible. 7 a Construct three particular sets, and C such that, C and C. b Construct three particular sets, and C for which, C and C. 8 a Construct two particular sets and such that and. b Construct two particular sets and such that and. Material from ICE-EM Mathematics, 3, 4. rown, Evans et al (2006/7)

12 D Sets and the number line Whole numbers The whole numbers are the numbers 0,, 2, 3,. These are often called the counting numbers, because they are the numbers we use when counting things. In particular, we have been using these numbers to count the number of elements of finite sets. Zero is a whole number, because zero is the number of elements of the empty set. The set of all whole numbers can be represented by dots on the number line ny set of whole numbers can be represented on the number line. For example, here is the set {0,, 4}: Sets and the number line The set of whole numbers is infinite. The set of whole numbers can be represented on the number line. Exercise D Write down, using set notation, the set represented by the blue dots. a b c d Represent each set on a number line. a {, 2, 5} b {0,, 5, 6} c {2, 3} d {4} Material from ICE-EM Mathematics, 3, 4. rown, Evans et al (2006/7) 2

13 e {whole numbers less than 5} f {even whole numbers less than 5 units from 6} g {odd whole numbers between 0 and 8} h the set of prime numbers less than 2 i the set of numbers that are 2 units from 3 j {even whole numbers between 3 and 9} k {whole numbers between and 4 inclusive} l {odd whole numbers between 3 2 and 4 2 } 3 List the elements of the set such that {whole numbers}, = 7, and the largest number in is 6. E Union and intersection We often need to talk about the overlap between sets. We use some new words and new notation to describe the situation. The intersection of two sets The intersection of two sets and consists of all elements belonging both to and to. This is written as. For example, some musicians are singers and some play an instrument. If = {singers} and = {instrumentalists}, then = {singers who play an instrument}. Here is an example using small sets of letters. If V = {vowels} and F = {letters in dingo }, then V F = {i, o}. This can be represented on a Venn diagram as follows. E V e a u i o d n g F 3 Material from ICE-EM Mathematics, 3, 4. rown, Evans et al (2006/7)

14 We return to a discussion of intersection and Venn diagrams in the next section. Intersection and the word and The word and tells us that there is an intersection of two sets. For example: {singers } {instrumentalists} = {people who sing and play an instrument} {vowels} {letters of dingo } = {letters that are vowels and are in dingo } The union of two sets The union of two sets and consists of all elements belonging to or to. This is written as. Note that the elements may belong to both. Continuing with the example of singers and instrumentalists: If = {singers} and = {instrumentalists}, then = {musical performers}. In the case of the sets of letters: If V = {vowels} and F = {letters in dingo }, then V F = {a, e, i, o, u, d, n, g}. This can be represented on a Venn diagram as follows. E V e a u i o d n g F We return to a discussion of union and Venn diagrams in the next section. Union and the word or The word or tells us that there is a union of two sets. For example: {singers } {instrumentalists} = {people who sing or play an instrument} {vowels} {letters in dingo } = {letters that are vowels or are in dingo } Material from ICE-EM Mathematics, 3, 4. rown, Evans et al (2006/7) 4

15 Note that the word or in mathematics always means and/or, so there is no need to add or both to these descriptions of the unions. For example, let = {even whole numbers that are less than 5} and = {whole numbers divisible by 3 that are less than 5}. Then: = {0, 2, 4, 6, 8, 0, 2, 4} and = {0, 3, 6, 9, 2} = {0, 2, 3, 4, 6, 8, 9, 0, 2, 4} Note: 6 and 2 are in both and. Disjoint sets Two sets are called disjoint sets if they have no elements in common. For example: The sets M = {men} and W = {women} are disjoint. The sets S = {2, 4, 6, 8} and T = {, 3, 5, 7} are disjoint. E S T nother way to define disjoint sets is to say that their intersection is the empty set: Two sets and are called disjoint if =. In the two examples above, M W = because no person is both a man and a woman, and S T = because no number lies in both sets. Union and intersection Let and be two sets. The intersection is the set of all elements belonging to and to. The union is the set of all elements belonging to or to. In mathematics, the word or always means and/or. The two sets are called disjoint if they have no elements in common, that is, if =. 5 Material from ICE-EM Mathematics, 3, 4. rown, Evans et al (2006/7)

16 Exercise E Use the words and and or to describe the sets P Q and P Q. a P = {boys who play cricket}, Q = {boys who play tennis} b P = {girls who study mathematics}, Q = {girls who study history} c P = {triangles with a right angle}, Q = {triangles with two equal sides} d P = {perfect square}, Q = {even whole numbers} e P = {animals with fur}, Q = {animals with webbed feet} 2 In each part, describe the sets and. Then state whether or not the sets are disjoint. a = {girls who play winter sport}, = {girls who play summer sport} b = {popular singers}, = {classical singers} c = {hard-boiled eggs}, = {soft-boiled eggs} d = {people born in ustralia}, = {people born overseas} 3 In each part, list the sets X Y and X Y, then state whether or not the sets are disjoint. a X = {0,, 5, 8}, Y = {0, 5, 8, 0, 2} b X = {0,, 4, 9, 6}, Y = {2, 3, 5, 7,, 3} c X = {a, b, c, d, e, f}, Y = {b, c, f, g, k} d X = {2, 4, 6, 8}, Y = {, 3, 5} e X = {, 2, 3,, 20}, Y = {5, 6, 7,, 50} f X = {2, 5, 0, 2, 4}, Y = {5, 0, 4} 4 In each part, list the elements of and, then represent them on separate number lines. a = {0, 3, 6}, = {, 2, 3} b = {odd numbers between 0 and 0}, = {whole numbers less than 6}. Material from ICE-EM Mathematics, 3, 4. rown, Evans et al (2006/7) 6

17 5 In each part, list the elements of C and C. a = {0, 4, 8, 2, 6}, = {0, 3, 6, 9, 2, 5}, C = {0, 0, 20} b = {, 3, 5, 7, 9, }, = {, 2, 3, 4, 5, 6}, C = {7, 8, 9, 0,, 2} 6 a Let = {mammals}. i Write down a subset S of. ii Write down another subset T of so that S and T are disjoint. b Let N = {whole numbers}. i Write down an infinite subset S of N. ii Write down another infinite subset T of N so that S and T are disjoint. 7 Explain why the following statements are true, where and are any two sets. a b c d e f 8 Copy and complete the following statements, where and can be any two sets. a If then = b If then = c = d = e = f = 9 Suppose and are finite sets. re the following statements true or false? Explain your answers. a = b = 0 c = + d = 7 Material from ICE-EM Mathematics, 3, 4. rown, Evans et al (2006/7)

18 F Set complements and Venn diagrams Complement of a set Suppose that a suitable universal set E has been chosen. The complement of a set S is the set of all elements of E that are not in S. The complement of S is written as S c. For example: If E = {letters} and V = {vowels}, then V c = {consonants} If E = {whole numbers} and O = {odd whole numbers}, then O c = {even whole numbers} Complement and the adverb not The adverb not always corresponds to the complement of a set. For example: If E = {letters} and V = {vowels}, then V c = {letters that are not vowels} = {consonants} If E = {whole numbers} and O = {odd whole numbers}, then O c = {whole numbers that are not odd} = {even whole numbers} Venn diagrams and set complements Note: In the next three Venn diagrams below, the universal set is E = {0,, 2, 3, 4, 5, 6, 7, 8, 9, 0}, and all these numbers are placed somewhere within the rectangle. In the first Venn diagram below, the region inside the circle represents the set of odd whole numbers between 0 and 0. Thus we place the numbers, 3, 5, 7 and 9 inside the circle, because = {, 3, 5, 7, 9}. Outside the circle, we place the other numbers in E (0, 2, 4, 6, 8 and 0) that are not in. Thus the region outside the circle represents the complement: c = {0, 2, 4, 6, 8, 0} E Material from ICE-EM Mathematics, 3, 4. rown, Evans et al (2006/7) 8

19 Venn diagrams of intersections and unions The Venn diagram below shows two sets and, where = {, 3, 5, 7, 9} and = {, 2, 3, 4, 5}. The numbers, 3 and 5 lie in both sets, so we place them in the overlapping region of the two circles. The remaining numbers in are 7 and 9; these are placed inside but outside. The remaining numbers in are 2 and 4; these are placed inside but outside. E Thus the overlapping region represents the intersection = {, 3, 5}, and the two circles together represent the union, because = {, 2, 3, 4, 5, 7, 9}. The four remaining numbers 0, 6, 8 and 0 are placed outside both circles. Representing disjoint sets on a Venn diagram When we know that two sets are disjoint, we represent them by circles that do not intersect. For example, let P = {0,, 2, 3} and Q = {8, 9, 0}. Then P and Q are disjoint, as illustrated in the Venn diagram below. E 5 P Q Material from ICE-EM Mathematics, 3, 4. rown, Evans et al (2006/7)

20 Set complements and Venn diagrams We often choose a convenient universal set that contains all the elements under discussion. The complement of a particular set consists of all elements of the universal set that are not in the set under discussion. Sets are represented in a Venn diagram by circles inside the universal set. The overlapping region of two circles represents the intersection of the two sets, and the two circles together represent their union. When two sets are known to be disjoint, the circles are drawn without any overlap. When one set is known to be a subset of another, its circle is drawn inside the circle of the other set. Exercise F In all of the following, E stands for the universal set. Write down the complement of each set S in the given universal set E. a S ={women}, where E = {adults} b S = {boys}, where E = {children} c S = {ustralians living within 0 km of the coast}, where E = {ustralians} d S = {weekdays}, where E = {days} e S = {whole numbers less than 00}, where E = {whole numbers} f S = {prime numbers}, where E = {whole numbers} 2 The diagram below represents the set inside the universal set E. E List the elements of: a b E c c d c e c f E c g E h E i ( c ) c Material from ICE-EM Mathematics, 3, 4. rown, Evans et al (2006/7) 20

21 3 State whether each statement is true or false. Explain your answers. a ( c ) c = b c = E c c = E d E c = E e c = E f = c 4 From the Venn diagram below, list all the elements of the following sets. E c a b e g d f a E c e b d f {elements of that are not in } g {elements of that are not in } 5 From the Venn diagram below, list all the elements of the following sets. E P Q a P c P Q e P c g (P Q) c i P Q c b Q d P Q f Q c h (P Q) c j P Q c k Q P c l Q P c 6 Draw a Venn diagram to illustrate each situation, where and are subsets of the universal set E. a b c and are disjoint. d and have some common elements but neither set is a subset of the other. 7 For each part, draw a Venn diagram, then list and. a = {, 2, 4}, = {3, 5} b = {a, e, i}, = {e, i, o, s, t} c = {, 2, 3, 5, 8, 0}, = {3, 8, 0} 2 Material from ICE-EM Mathematics, 3, 4. rown, Evans et al (2006/7)

22 G Keeping count of elements of sets efore solving problems with Venn diagrams, we need to work out how to keep count of the elements of overlapping sets. E E (3) (3) (4) (2) The left-hand diagram above shows two sets and inside the universal set E, where = 6 and = 7, with 3 elements in the intersection. The right-hand diagram shows only the number of elements in each of the four regions. These numbers are placed inside round brackets so that they don t look like elements. Notice carefully that = 6 and = 7, but The reason for this is that the overlapping region should only be counted once, not twice. When we subtract the three elements of from the total, the calculation is then correct: = = 0 Example In the diagram shown below, = 5, = 25, = 5 and E = 50. E a Insert the number of elements into each of the four regions. b Hence find and c. (continued on next page) Material from ICE-EM Mathematics, 3, 4. rown, Evans et al (2006/7) 22

23 Solution E (0) (5) (20) (5) a The intersection has 5 elements. Hence the region of outside must have 0 elements, and the region of outside must have 20 elements. This makes 35 elements so far, so the outer region has 5. b Hence = 35 and c = 0. Example 2 In the diagram shown below, S = 5, T = 20, S T = 25 and E = 50. E S T a Insert the number of elements into each of the four regions. b Hence find S T and S T c. 23 Material from ICE-EM Mathematics, 3, 4. rown, Evans et al (2006/7)

24 Solution E S T (5) (0) (0) (25) a The union S T has 25 elements, whereas S has 5 elements and T has 20 elements, so the overlap S T must have 0 elements. Hence the region of S outside S T must have 5 elements, and the region of T outside S T must have 0 elements. Finally, the region (S T ) c inside E but outside S T must have = 25 elements. b Hence S T = 0 and S T c = 40. Number of elements in the union of two sets The number of elements in the union of two sets and can be found by: Number of elements in = number of elements in + number of elements in number of elements in Exercise G = + The sets and are subsets of the universal set E = {, 2, 3, 4, 5, 6, 7, 8, 9, 0}. For each question part, draw a Venn diagram with two overlapping sets and insert in each region the number of elements, not the elements themselves. Then write down and. a = {, 2, 3, 4}, = {2, 3, 4, 5} b = {, 2, 3, 4, 5}, = {2, 4, 6, 8, 0} c = {5, 7, 8, 0}, = {, 3, 4} Material from ICE-EM Mathematics, 3, 4. rown, Evans et al (2006/7) 24

25 Example,2 2 Draw Venn diagrams to solve the following problems. a Find if = 4, = and = 6. b Find if = 25, = 38 and = 3. c Find if = 25, = 2 and = 0. d Find if = 7, = 8 and = 0. e Find if = 0, = 5 and = 2. f Find if = 6, = 20 and = a pair of dice was rolled 00 times. The sum of the numbers on the upturned faces was greater than 6 in 70 rolls, and less than 8 in 65 rolls. Draw a Venn diagram and find how many times this sum was 7. b One hundred boys all play cricket or soccer. Eighty play cricket and 67 play soccer. Draw a Venn diagram and find out how many play both sports. c One hundred girls each wrote down the value of the coins in their pockets. Sixty of them had more than $ and 48 of them had less than $.20. Draw a Venn diagram to find how many of them had $.05, $.0 or $.5. d mong 00 concert-goers, 75 are singers, 65 are instrumentalists and 50 are both singers and instrumentalists. Draw a Venn diagram to find out how many are neither singers nor instrumentalists. e coin collector buys a bag containing 00 coins. Sixty-five are silver, 90 are round, and 6 are silver and round. Draw a Venn diagram to find out how many are neither silver nor round. f One hundred and sixty students all play the piano or the violin. One hundred and twenty play the piano and 80 play the violin. How many play both? 25 Material from ICE-EM Mathematics, 3, 4. rown, Evans et al (2006/7)

26 H Problem solving using Venn diagrams This last section explains how to use Venn diagrams to solve counting problems. The approach is to identify the sets involved, and then construct a Venn diagram to keep track of the numbers in the different regions of the diagram. Example 3 travel agent surveyed 00 people to find out how many of them had visited the cities of Melbourne and risbane. Thirty-one people had visited Melbourne, 26 people had been to risbane, and 2 people had visited both cities. Draw a Venn diagram to find the number of people who had visited: a Melbourne or risbane c only one of the two cities b risbane but not Melbourne d neither city. Solution Let M represent the set of people who had visited Melbourne, let be the set of people who had visited risbane, and let the universal set E be the set of people surveyed. Then the information given in the question can be rewritten as M = 3, = 26, M = 2, E = 00. Hence number in M only = 3 2 = 9 and number in only = 26 2 = 4. E M (9) (2) (4) (55) (continued on next page) Material from ICE-EM Mathematics, 3, 4. rown, Evans et al (2006/7) 26

27 a Number visiting Melbourne or risbane = = 45 b Number visiting risbane only = 4 c Number visiting only one city = = 33 d Number visiting neither city = = 55 Problem solving using Venn diagrams First identify the sets involved. Then construct a Venn diagram to keep track of the numbers in the different regions of the diagram. Exercise H Solve each of the following problems by drawing a Venn diagram. Example 3 oys in a class play tennis or swim or both. Fifteen play tennis, 0 swim and 5 both play tennis and swim. a How many boys play tennis but do not swim? b How many boys swim but do not play tennis? c How many boys play tennis or swim but do not do both? d How many boys are in the class? 2 Of a group of 80 people on a railway station, 48 carry umbrellas and 52 wear raincoats, while 40 both carry an umbrella and wear a raincoat. a How many have an umbrella, but no raincoat? b How many have a raincoat, but no umbrella? c How many have an umbrella or raincoat or both? d How many have neither raincoat nor umbrella? 27 Material from ICE-EM Mathematics, 3, 4. rown, Evans et al (2006/7)

28 3 The Royal Zoo has 500 animals, of which 400 can walk, 50 can swim, and 30 can both walk and swim. a How many can walk, but not swim? b How many can swim, but not walk? c How many can walk or swim? d How many can neither walk nor swim? 4 The Republican Zoo has a collection of 70 interesting snakes. Fiftyfive of them are poisonous, 45 are more than m long, and 67 are either poisonous or more than m long. a How many are poisonous and more than m long? b How many are poisonous, but m or less long? c How many are more than m long, but not poisonous? d How many are neither poisonous nor more than m long? 5 Two hundred and forty university students take part in a demonstration, all of them either carrying banners or sitting down on the road or both. Eighty-four carry banners and 204 sit on the road. Find how many students: a sit down on the road and carry banners b do only one of these things. 6 In a factory, 000 light bulbs were subjected to two quality control tests and. Five hundred and sixty passed test and 40 passed test, while 50 passed both tests. How many light bulbs: a failed both tests? b passed test but not? c passed test but not? d passed only one of the tests? e failed at least one of the tests? f failed in only one of the tests? Material from ICE-EM Mathematics, 3, 4. rown, Evans et al (2006/7) 28

29 7 4th-century ook of Saints contains the names of 300 men and 200 women. There are 250 martyrs amongst the men and 80 amongst the women; 20 of the men and 05 of the women healed the sick; and 200 of the men and 00 of the women both healed the sick and were martyrs. a How many saints were martyrs, but didn t heal the sick? b How many saints healed the sick, but were not martyrs? c How many saints neither healed the sick nor were martyrs? d How many of the women saints neither healed the sick nor were martyrs? 8 group of 00 children was asked which of the three films Hook, atman and Dracula they had seen. Twenty-seven had seen Hook, 38 had seen atman and 6 had seen Dracula. Eleven children had seen both Hook and atman, 8 had seen both Hook and Dracula, and 6 had seen atman and Dracula, while 3 had seen all three films. Find how many of the children had seen: a at least one of the three films c only one of the films e exactly two of the three films. b Hook only d none of the films 9 In a certain school, there are 80 pupils in Year 7. One hundred and ten pupils study French, 88 study German and 65 study Indonesian. Forty pupils study both French and German, 38 study German and Indonesian, and 26 study both French and Indonesian, while 9 study German only. Find the number of pupils who study: a all three languages c none of the languages b Indonesian only e either one or two of the three languages. d at least one language 29 Material from ICE-EM Mathematics, 3, 4. rown, Evans et al (2006/7)

30 20C Sets and Venn diagrams Exercise a {, 3, 5, 7, 9, } b {3, 23, 33, 43, 53, 63, 73, 83, 93} c {January, February, May, July} d {sia, ustralia, frica, ntarctica, Europe, North merica, South merica} e {2, 8, 24, 30, 36} f {4, 2, 28, 35} g {, 4, 9, 6, 25, 36, 49} h {2, 3, 5, 7,, 3, 7, 9, 23, 29} 2 a {first 3 letters of the English alphabet} b {odd numbers less than 6} c {the first 4 prime numbers} Other answers are possible. 3 a no ( best depends on people s opinions) b yes c yes d no (there is no consensus on bright colours) e no (there is no consensus on small ) g no (some people change their votes with time) f yes 4 and D 5 a = b = c d = e (not everyone is a citizen) 6 a 9 S b 2 S c 5 S d 0 S 7 a b c d e f g h Exercise a not defined b finite c infinite d finite e infinite f finite g finite 2 a 3 b 6 c 20 d 26 e 6 f 8 g 4 h 2 i 0 j k 2 l 5 3 a {0,, 2,, 9}, finite b {, 3, 5,, 9}, finite c {, 4, 9, }, infinite d {5, 52, 53, }, infinite e {2, 4, 6}, finite g {2, 3, 5, 7,, 3, 7, 9, 23, 29}, finite f {56, 63, 70, }, infinite 4 a empty b non-empty c empty d empty e non-empty f empty 5 a false b true c false d true e false f true 6 a = b c = d = e Exercise C f = g h i = j = Material from ICE-EM Mathematics, 3, 4. rown, Evans et al (2006/7) 30

31 Exercise C a b c d e f 2 a b c d e f g h i j k 3 a Everything in belongs to S. b Everything in belongs to, and everything in belongs to C, so everything in belongs to C. c Everything in S is in S. d ny element of belongs to and vice versa, so and must be equal. e Everything in {x} is in S. f Everything is {x, y} is in S. g ll elements of are in, so must have at least as many elements as. 4 a False; could be equal to. b False; they can have the same number of elements. c True; they have the same elements. d True; C has all the elements of and maybe more. 5 a i {a, b, c}, {a, b}, {b, c}, {a, c}, {a}, {b}, {c}, ii {a, b}, {a}, {b}, iii {a}, b The number is 2 to the power of the number of elements. 6 a {fish} {animals} b {numbers ending in 6} {even numbers} c {children with blonde hair} {naughty people} d {numbers ending in 4} {multiples of 4} 7 nswers may vary. Examples are: e {adults} {sensible people} a = {}, = {, 2}, C = {, 2, 3} b = {}, = {, 2, 3}, C = {, 2} 8 nswers may vary. Examples are: a = {, 2}, = {} b = {, 2}, = {, 3} 3 Material from ICE-EM Mathematics, 3, 4. rown, Evans et al (2006/7)

32 Exercise D a {, 2, 4, 6} b {0, 2, 3, 5, 7} c {2, 3, 4, 5} d {2} 2 a b c d e f g h i j k l {0,, 2, 3, 4, 5, 6} Exercise E a P Q = {boys who play cricket and tennis}; P Q = {boys who play cricket or tennis} b P Q = {girls who study mathematics and history}; P Q = {girls who study mathematics or history} c P Q = {right isosceles triangles}; P Q = {triangles which are right or isosceles triangles} d P Q = {even perfect square numbers}; P Q = {even numbers or perfect square numbers} e P Q = {animals with fur and webbed feet}; P Q = {animals with fur or webbed feet} 2 a = {girls who play winter and summer sport}; = {girls who play winter or summer sport}; not disjoint b = {singers who sing popular and classical music}; = {singers who sing popular or classical music}; not disjoint Material from ICE-EM Mathematics, 3, 4. rown, Evans et al (2006/7) 32

33 c = ; = {hard- or soft-boiled eggs}; disjoint d = ; = {all people}; disjoint 3 a X Y = {0, 5, 8}; X Y = {0,, 5, 8, 0, 2}; not disjoint b X Y = ; X Y = {0,, 2, 3, 4, 5, 7, 9,, 3, 6}; disjoint c X Y = {b, c, f}; X Y = {a, b, c, d, e, f, g, k}; not disjoint d X Y = ; X Y = {, 2, 3, 4, 5, 6, 8}; disjoint e X Y = {5, 6, 7, 8, 9, 20}; X Y = {, 2, 3,, 50}; not disjoint f X Y = {5, 0, 4}; X Y = {2, 5, 0, 2, 4}; not disjoint 4 a = {3} = {0,, 2, 3, 6} b = {, 3, 5} = {0,, 2, 3, 4, 5, 7, 9} a C = {0}; C = {0, 3, 4, 6, 8, 9, 0, 2, 5, 6, 20} b C = ; C = {, 2, 3, 4, 5, 6, 7, 8, 9, 0,, 2} 6 nswers may vary. For example: a i S = {rodents} b i S = {even whole numbers} ii T = {primates} ii T = {odd whole numbers} 7 a nything in the intersection of and must be in. b nything in the intersection of and must be in. c The union of with anything must contain. d Elements common to and belong to the union of and. e The empty set is a subset of any set. f s for e 8 a b c d e f 9 a True; =. b True; =. Exercise F c,d False; counter-examples are readily constructed. 33 Material from ICE-EM Mathematics, 3, 4. rown, Evans et al (2006/7)

34 Exercise F a {men} b {girls} c {ustralians living more than 0 km away from the coast} d {Saturday, Sunday} e {whole number greater than or equal to 00} f {0,, and composite numbers} 2 a {5, 6} b {0,, 2, 3, 4, 5, 6} c {0,, 2, 3, 4} d {0,, 2, 3, 4, 5, 6} e f g {5, 6} h {0,, 2, 3, 4, 5, 6} i {5, 6} 3 a True; not not in is the same as in. b True; something is either in or not in. c False; the right hand side should be. d False; the right hand side should be. e True; the universal set is everything. f False; counter examples are easily constructed, the size of c is the size of E the size of. 4 a {a, b, c, d, e, f, g} b {a, b, e} c {b, d, e, g} d {b, e} e {a, b, d, e, g} f {a} g {d, g} 5 a {, 3, 5, 7, 8} b {, 2, 3, 6, 9, } c {, 3} d {, 2, 3, 5, 6, 7, 8, 9, } e {2, 4, 6, 9, 0,, 2} f {4, 5, 7, 8, 0, 2} g {4, 0, 2} h {2, 4, 5, 6, 7, 8, 9, 0,, 2} i {5, 7, 8} j {, 3, 4, 5, 7, 8, 0, 2} k {2, 6, 9, } l {, 2, 3, 4, 6, 9, 0,, 2} 6 a E b E c E d E 7 a E b E a e i o t s = {, 2, 3, 4, 5}, = = {a, e, i, o, s, t}, = {e, i} Material from ICE-EM Mathematics, 3, 4. rown, Evans et al (2006/7) 34

35 c E Exercise G = {, 2, 3, 5, 8, 0}, = {3, 8, 0} a E b E () (3) () (3) (2) (3) (5) (2) c E = 5, = 3 = 8, = 2 (4) (0) (3) (3) = 7, = 0 2 a 9 b 50 c 46 d 5 e 4 f 0 3 a 35 b 47 c 8 d 0 e 6 f 40 Exercise H a 0 b 5 c 5 d 20 2 a 8 b 2 c 60 d 20 3 a 270 b 20 c 420 d 80 4 a 33 b 22 c 2 d 3 5 a 48 b 92 6 a 80 b 40 c 260 d 670 e 850 f a 30 b 5 c 55 d 5 8 a 59 b c 40 d 4 e 6 9 a 9 b 0 c 2 d 68 e Material from ICE-EM Mathematics, 3, 4. rown, Evans et al (2006/7)

36 2C Further uses of Venn diagrams Sometimes, instead of listing the outcomes of the experiment in the appropriate events on a Venn diagram, the number of outcomes in each event is written on the diagram. Consider the following examples. Example 7 In a group of 25 families, 4 own a Playstation, 5 own an Xbox, and 6 own both a Playstation and an Xbox. Represent this information on a Venn diagram. Solution Let be the event family owns a Playstation and be the event family owns an Xbox. There are: 6 elements in 4 6 = 8 elements in c 5 6 = 9 elements in c 2 elements in c c since = 25 X Example 8 In a class of 28 students, 20 study French and 5 study Chemistry. Each student in the class studies either French or Chemistry. a Represent this information on a Venn diagram. b One student is selected at random from the group. What is the probability that the student studies: i both French and Chemistry ii Chemistry but not French? Material from ICE-EM Mathematics, 3, 4. rown, Evans et al (2006/7) 36

37 Solution a Let F be the event student studies French and C be the event student studies Chemistry. Twenty students study French and fifteen study Chemistry in a class of 28. F C = F + C F C 28 = F C X F C Therefore F C = Then 20 7 = 3 students study French but not Chemistry, and 5 7 = 8 students study Chemistry but not French. 0 b i Using the Venn diagram, P(F C) = 7 28 = 4 ii Using the Venn diagram, P(F c C) = 8 28 = 2 7 Exercise 2C Example 7 In a group of 40 children, 2 like both heavy metal and rap; 8 children like heavy metal but not rap; 4 children like rap but not heavy metal; and the rest like neither. Represent this information on a Venn diagram. 2 survey of 50 families, showed that 28 owned a cat, 30 owned a dog and 5 owned both a dog and a cat. a Represent this information on a Venn diagram. b One family is selected at random from the group. What is the probability that the family owns: i a cat but not a dog ii a dog but not a cat iii neither a dog nor a cat? 37 Material from ICE-EM Mathematics, 3, 4. rown, Evans et al (2006/7)

38 Example 8 3 In a class of 26 students, 8 study Chemistry and 2 study Economics. Each student studies either Chemistry or Economics. a Represent this information on a Venn diagram. b One student is selected at random. What is the probability that the student studies: i Chemistry and Economics ii only Economics iii only Chemistry? 4 In a youth club of 50 people, 8 play basketball, 28 play soccer and 0 play neither sport. a Represent this information on a Venn diagram. b One person is selected at random. What is the probability that the person chosen plays: i basketball and soccer ii soccer but not basketball iii only basketball? 5 In a group of 50 students, 30 study Mathematics, 25 study Physics and 20 study both. a Represent this information on a Venn diagram. b One student is selected at random from the group. What is the probability that the student studies: i Mathematics but not Physics ii Physics but not Mathematics iii neither Physics nor Mathematics? 6 In a group of 00 students, 50 study History, 30 study English Literature and 20 study both. a Represent this information on a Venn diagram. b If a student is selected at random from the group, what is the probability that the student studies: i at least one of these subjects Material from ICE-EM Mathematics, 3, 4. rown, Evans et al (2006/7) 38

39 ii History but not English Literature iii History, given that the student also studies English Literature? 7 Use Venn diagrams to simplify each of the following events or express in a different form. a ( c ) ( ) b ( ) c c ( ) c d ( c ) ( c ) e (( c c ) ( c )) c 39 Material from ICE-EM Mathematics, 3, 4. rown, Evans et al (2006/7)

40 Chapter 2 answers Exercise a 3 0 a b 2 b c 4 c d a b a There are 36 equally likely outcomes. total of 7 happens with (, 6), (6, ), (2, 5), (5, 2), (3, 4) and (4, 3). So the probability of getting a 7 is 6 36 = 6. b There are 4 equally likely outcomes, (H, T), (H, H), (T, H) and (T, T). So the probability of tossing two heads is 4. 7 c a 27 a Exercise b 4 b a 0 c 3 d 4 3 c b 0 26a 5 22 d 0 28a c b 22 b d 0 c 22 2 c 5 d d 3 0 X X X a 2 b 2 5 c 3 0 d 7 0 e 7 0 f 5 g 0 5 a 2 b 6 c 2 d 6 e 2 f 0 g 2 3 h 6 Exercise 2C i X M R a X C D 7 b i ii iii a X C E 0 2 b i 3 4 ii 3 7 iii 3 Material from ICE-EM Mathematics, 3, 4. rown, Evans et al (2006/7) 40

41 4 a X S a X H E b i b i ii iii ii a X M P 7a c c c b c c b i ii iii iii 2 3 d Ø e 4 Material from ICE-EM Mathematics, 3, 4. rown, Evans et al (2006/7)

42 3E Inclusion-exclusion principle In earlier sections, we have used the multiplication principle to count the number of arrangements of sets of objects with and without repetition. This section concentrates on counting the elements in subsets of a set X so we shall begin by revising some basic ideas from set theory. The Venn diagram was introduced in Chapter 2 of ICE-EM Mathematics Secondary 3. It is the standard way of representing a set. be the subset of the set X then the set c, called the complement of, is the set of all elements in X which are not in. X c Two subsets, of X split the Venn diagram up into 4 regions. The set of elements of X which belong to both and is called the intersection of and, and is denoted by. X Clearly the 4 regions can all be described using complements and intersections. X c c c c We use to mean the number of elements in the set. So X = + c and X = c c + c + c + The other standard construction is the union of and. This is all the elements of X which belong to or (or both and ). X Material from ICE-EM Mathematics, 3, 4. rown, Evans et al (2006/7) 42

43 Example 4 How many numbers between and 20 are a multiple of both 2 and 3? How many numbers between and 20 are a multiple of either 2 or 3? Solution X = {, 2, 3,, 20}, then X = 20. be the numbers in X which are multiples of 2. Then = {2, 4, 6,, 20} and = 0. be the numbers in X which are multiples of 3. Then = {3, 6, 9,, 8} and = 6. are the numbers in X which are multiples of 2 and 3, that is the multiples of 6. = {6, 2, 8}. so = 3. is not + since this counts the numbers in twice. So = + = = 3 In summary: The number of multiples of 2 and 3 between and 20 is 3. The number of multiples of 2 or 3 between and 20 is 3. Notes:. We can draw a Venn diagram presenting the information in Example 4 as follows: From this we can deduce, for example, ( ) c = 7. That is there are 7 numbers between and 20 which are neither a multiple of 2 nor a multiple of 3. X Material from ICE-EM Mathematics, 3, 4. rown, Evans et al (2006/7)

44 2. If X is much larger, say X = 00, then listing all elements is impractical. We use another type of Venn diagram where the number of elements in each region is recorded, not the actual elements. So for Example 4 we have: Now = is easy to read off. We shall use this type of Venn diagram from now on. 3. The formula = + is called the inclusion-exclusion principle for 2 subsets of X. It is clearly true for 2 subsets of any finite set X since + counts the elements of except that is counted twice. 4. If is empty, = then = +. and are said to be disjoint subsets of X. Example 5 In a group of 40 students, 30 study German (G) and 20 study French (F). Five students study neither language. How many students study a French and German Solution X X b German only? a F G = 40 5 = 35 F G = F + G F G 35 = F G F G = 5 X F Hence 5 students study both languages. b 30 5 = 5 students study German only. 5 G Note: F G is the set of students who study French and German F G is the set of students who study French or German. So intersection is closely related to and and union is closely related to or. Chapter 3 Combinatorics 07 Material from ICE-EM Mathematics, 3, 4. rown, Evans et al (2006/7) 44

45 Inclusion-exclusion for three subsets Suppose, and C are three subsets of the set X. Then the standard Venn diagram is: X There are eight subsets of X determined by, and C, C is the central region in the diagram and C is all of X except the outer region. C The inclusion-exclusion principle for 3 subsets gives the formula for C in terms of,, C,, C, C and C. Clearly we must begin by adding, and C. t this stage, we have counted all the elements in C but we have counted all of the elements in the sets, C and C twice. Hence we must subtract, C and C. Finally, we must add C since it has been counted 3 times and subtracted 3 times. In summary: C = + + C C C + C 45 Material from ICE-EM Mathematics, 3, 4. rown, Evans et al (2006/7)

46 Example 7 a How many numbers between and 000 are divisible by 7, or 3? b How many numbers between and 000 are not divisible by 7, or 3? Solution a X = {, 2, 3,, 000}. be the set of numbers in X divisible by 7. = 42 since 000 = be the set of numbers in X divisible by. = 90 since 000 = C be the set of numbers in X divisible by 3. C = 76 since 000 = Next, is the set of numbers divisible by both 7 and so = 2 since 000 = Similarly, C = 0 since 000 = and C = 6 since 000 = Finally, C = since 7 3 = 00 > 000. Next we use the inclusion-exclusion principle. C = + + C C C + C = = 280 (continued on next page) Since X = ( C) ( C) c so ( C) c = 720. So we have shown that there are 280 numbers between and 000 divisible by 7, or 3. b There are 720 numbers between and 000 not divisible by either 7, or 3. Note: If we draw a Venn diagram including the size of the 8 regions then we can check our answers. X X = C Material from ICE-EM Mathematics, 3, 4. rown, Evans et al (2006/7) 46