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1 SET THEORY I. Basic Notions A set is a collection of objects. The objects in the set are referred to as elements. A set can be finite, meaning that there is a limited number of objects in the set or infinite, meaning that there is no end to the number of objects in the set. We usually use a capital letter to designate a set. There are two ways that we can specify a set. One way is called a roster, in which we list the elements of the set in set brackets: A = { a,b,c,d,e} or M = { 1,2,3,…9,10} or C = { 1,2,3,4,…} The “…” part of set M does not mean it is infinite; it simply means that a pattern has been established and the reader can fill in the missing numbers. The “…” portion of set C, however, does mean that it is infinite, no numbers are specified to end the list of numbers. All three of these sets have been designated by roster. The other way to specify a set is using set-builder notation. This notation includes the set brackets, a variable to indicate a member of the set and the a vertical bar (read as “such that”) and then a description of the elements of the set. Examples are: B = { x| x is a real number and -2 < x < 4 } or D = { y| y is a state in the USA} Set B is an infinite set and D is a finite set but both are specified by set-builder notation. For a set to be useful, it must be well-defined. For instance, sets B and D above are well-defined. It is easy for us to determine if a particular object is a member of the set. An example of a set that is not well defined would be {x| x is Susie’s friend}—we do not know how well a person needs to know Susie to be classified as a “friend.” There is a special symbol used to indicate the relationship “is an element of”. The following statement is read “ a is an element of A” a A I. EXERCISES Find the elements in each set. List all elements in a roster. 1) { 11, 12, 13, …,16} 2) { 8,9,10,…16} 3) { x| x is a counting number less than 7} 4) { x| x is a counting number greater than 4 but less than 10} 5) { x| x is an integer and -3 ≤ x < 5} 6) { x| x is a vowel } 7) { 17, 22, 27, …, 47} 8) { 74, 68, 62, …, 38} 9) { x| x is an odd counting number between 8 and 16} 10) {x| x is an odd counting number not greater than 15} 11) {x| x is an even counting number less than 20} 12) {x| x is a counting number less than 100 and is evenly divisible by 21} 2 Identify each set as finite or infinite. 13) { 3,4,5,…8} 14) {1, ½ , ⅓, ¼, …} 15) {0, 1,2,3,…,75} 16) {x| x is a state in the United States} 17) {s| x is a real number } 18) {x| x is a letter in the English alphabet} 19) {x| x is an integer and -5 <x < 3} 20) {x| x is a rational number and 0 < x < 1} Identify each set as well-defined or not well-defined. 21) { 3,4,5} 22) { x| x is a member of this year’s football team at your school} 23) { 5, 10, 15, 20, …} 24) { x| x is a real number greater than 2} 25) { x| x is a rational number} 26) { x| x is a good movie} 27) { x| x is a lousy artist} 28) { x| x is a counting number less than 1} 29) { x| x is a mouse weighing over 500 pounds} Fill in the blank with either 30) 31) 32) 33) 34) 35) or to make a true statement. 3 ______ { x| x is an odd counting number} -4______ { x| x is an integer and |x| > 5} 0 ______ {x| x is a counting number } {3}_____ { 1, 2, 3, 4, 5 } -12_____{ 10, 12, 14, 16} {12}____{ 10,12,14,16} Write TRUE or FALSE for each of the following. 36)3 {1,2,3} 37)15 {0,5,10,15,20,...} 37)5 {0,2,4,6,...} 38)1 {7,5,3,1} 39){2} {0,2,4,6} 40) 1 {2,4,6,8} Write TRUE of FALSE for each statement, using the following sets: A = {2, 4, 6, 8, 10} B = { 2, 4, 8} C = { 4, 6, 12 } 41) 4 A 42)8 C 43)0 B 44) 2 C 45)6 A 3 II. Subsets For a set to be useful, it must be well-defined. For instance, sets B and D above are well-defined. It is easy for us to determine if a particular object is a member of the set. An example of a set that is not well defined would be {x| x is Susie’s friend}—we do not know how well a person needs to know Susie to be classified as a “friend.” There is a special symbol used to indicate the relationship “is an element of”. The following statement is read “ a is an element of A” a A Sets can have four relationships. They are: 1) A=B Equal sets contain exactly the same elements, in any order. 2) Sets A and B are disjoint. Disjoint sets have no elements in common. 3) Sets A and B overlap. This means that there are some elements the two sets have in common, but there are other elements that belong to just A and some that belong to just B. 4) A is a subset of B. This means that every element in A is also in B. The following statement is read “ A is a subset of B.” A B Every set is considered to be a subset of itself. Also, there is a special set, called the null or empty set, the set that does not contain any elements at all. This set is indicated by set brackets with nothing in between them, { }, or by a special symbol, Ø. The empty set is considered to be a subset of every set also. For each subset with n elements, there are 2n subsets. Another useful set to consider is the universal set, the set containing all the elements under current consideration. The universal set is usually indicated by the letter U. The set that is considered the universal set might change from problem to problem. Once a universal set is defined, it is possible to also define the complement of a set. The complement of set A, denoted as A', is the set of all elements in U that are not in A. For example if U = { 1,2,3,4,5} and A = { 1,3,5}, then A' = { 2,4}. II. EXERCISES Insert the symbol 1) 2) 3) 4) 5) 6) 7) 8) 9) or to make a true statement. {1,3}_______{1,3,5,7} {5}_______{x| x is an integer and 0 < x< 5} { } _______{ 6,7,8} {3,4}_______{3,4} {1,2}_______{x| x is a counting number} 4______{2,4,6,8,…} {4}______{2,3,4,5,…} { } _____{ } {3,4,5,6}________{x| x is an odd counting number} Identify each statement as TRUE or FALSE, using the following sets. 4 A= {2,4,6,8,10, 12} B={2,4,6} C ={4,10,12} D = {2,10} E = { 2,4,6,8,10,12,14} 10) A B 11) B A 12) B B 13){} A 14)C B 15) D A 16) D E 17)C D 18) D C 29) B E 20) There are exactly 8 subsets of C. 21) There are exactly 4 subsets of D. 22) There are exactly 6 subsets of B. 23) There is only 1 subset of Ø. Find the number of elements in each set. Then find the number of subsets for each set. 24) { 4,5,6} 25) { 6} 26) {0,1,2,3} 27) { } 28) {x| x is a counting number and 5 < x< 10} 29) {x| x is an integer and |x| ≤ 3} Let U = { 0,1,2,3,4,5,6,7,8}. Find the complement of each set. 30) {2,4,6} 31) {x| x is an even counting number less than 10} 32) {5,6,7,8} 33) {3} 34) { } 35) U Let C represent the set of symptoms for a cold. Then C = { sneezing, red nose, aches} Let F represent the set of symptoms for flu. Then F = {sneezing, red nose, sore throat, fever} Let U represent the set containing the combined symptoms for cold and flu. List the elements for the following sets: 36) U 37) C' 38) F' aches, 5 III. Set Operations There are two set operations that allow us to form new sets. Those operations are Union, in which we combine all members of both sets into a new set (but we do not repeat common elements) and intersection, in which we list the common elements for the two sets. Some examples may make this clear. We will use these symbols: means int er sec tion means union A = { 1,2,3,4,5} and B = { 2,4,6} and C = {1,2,3,4,5,6} A B {2,4} A C {1,2,3,4,5} A B {1,2,3,4,5,6} We can also combine the notion of complements with the idea of union and intersection. Let U = {1,2,3,4,5,6,7,8} and sets A, B and C be as above. Then A' B {6,7,8} {2,4,6} {6} ( A B)' {1,2,3,4,5,6}' {7,8} ( B C ) A' {2,4,6} {6,7,8} {2,4,6,7,8} III. EXERCISES Write TRUE or FALSE for each statement. 1){1,3,5,7} {5,6,7,8} {5,7} 2){2,1,5} {1,2,3} {1,2,3,4,5} 3){3,5,7} {1,2,3,4,5} {3,5} 4){} {8,7,6,5} {8,7,6,5} 5){} {8,7,6,5} {8,7,6,5} 6){1,3,4} {1,3,4} {} 7){1,3,4} {1,3,4} {1,3,4} 8){} {} {} 9){} {} {} 10){2,4} {3,4,5} {2,3,4,5} 6 For the next set of exercises, use these sets: U={3,4,5,6,7,8,9} X={3,4,5} Z={4,7,8,9} Y={3,5,7,8} List the elements in each set: 11) X Y 12) X Y 13)Y Z 14)Y Z 15) X U 16) Z U 17) X ' 18)Y 1' 19) X 'Y ' 20) X ' Z 21) Z '{} 22)Y '{} 23) X (Y Z ) 24)Y ( X Z ) 25)(Y Z ' ) X 26)( X 'Y '0 Z 27)( X Y )' Z ' 28)(Y ' X )' Z For the following problems, describe each set in words: A={all students over 17} B={all students studying Discrete Math} C={all students studying psychology} D={all students with brown eyes} U={all students} 29) A B 30) B C 31) A D 32) A C 33) A'C 34) A' B ' 35)( A') B C 36)( A B ) C ' 7 For the following statements, assume sets A and B are any two sets. Classify the statements as TRUE or FALSE. 37) A ( A B ) 38) A ( A B ) 39)( A B ) A 40)( A B ) A For the following exercises, let sets X, Y, Z and U be defined as in exercises 1-28. Find the set defined on the left hand side of the equals sign and then find the set defined on the right hand side. If the two are the same, write TRUE. If they are different, write FALSE. 41) X Y Y X 42)( X ' )' X 43) X {} {} 44)Y {} Y 45) Z {} Z 46)( X Y ) Z X (Y Z ) 47( X Y ) Z X (Y Z ) 8 IV. Venn Diagrams We can depict set relationships in pictures called Venn diagrams. In these pictures, the universal set is seen as a rectangle and two sets are seen as overlapping circles. We shade in the parts of the picture that represent the set under consideration. Consider the following pictures: IV. EXERCISES Place the elements of these sets in the proper location in the Venn diagram. 1) U=a,b,c,d,e,f,g} A={b,d,f,g} B={a,b,d,e,g} 2) U={5,6,7,8,9,10,11} A={5,8,9,10} B={7,8,9,10} Use a Venn diagram similar to the ones above to shade each of the following sets. 3) B A' 4) A B 5) A' B 6) A' B ' 7) B ' A 8) A' A 9)U ' 10)( A B ) B ' 11) A'( A B ) 9 Venn diagrams can also be used to depict relationships among three sets. 12)Use these sets to place the elements in the correct spot in the Venn diagram. U={1,2,3,4,5,6,7,8,9} A=(1,3,5,6} B=(2,4,6,8} C={2,3,4,5,6} Use a Venn diagram to shade each of the following sets. 13)( A B) C 14)( A B) C 15))( A' B) C 16)( A B)'C 17)( A'C ' ) B 18)( A B' ) C ' 19)( A C ' ) B' 20) B ( A C ' )' Venn diagrams can be used to solve counting problems. In the following problem, use the information given to count the number of people who fit into each region of a Venn diagram with three circles. Start with the most involved area (the intersection of all 3 sets). Don’t forget to draw the rectangle for the universal set. Suppose a survey produces the following results: 120 take French 150 take History 115 take Science 63 take French and History 70 take History and Science 78 take French and Science 38 take all three 6 students do not take any of these three courses 10 We can then answer questions like : a) How many students take only History? b) How many students take only French? c) How many students were in this survey? Use Venn diagrams to answer the following: 21) A survey of 150 students concerning their club memberships at school showed that 78 students were in a foreign language club 60 students were in an honor club 54 students were in a service club 25 students were in foreign language and honor clubs 30 students were in honor and service clubs 32 students were in service and foreign language clubs 11 students were in all 3 types of clubs a) b) c) d) How many students were only in an honor club? How many students were only in a foreign language club? How many students were only in a service club? How many students were not in any of these three clubs? 22) Athletes were questioned as to their participation in three sports. Here are the results: 34 played soccer 61 played basketball 50 played tennis 25 played soccer and tennis 30 played basketball and tennis 8 played soccer and tennis 6 played all three 9 played none of these sports a) b) c) d) How many athletes played only basketball? How many played only tennis? How many played only one sport? How many athletes were questioned? 11 A B A B A' A' B