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1、数据库系统基础教程第三章答案Exercise 3、1、1 Answers for this exercise may vary because of different interpretations、Some possible FDs: Social Security number name Area code state Street address, city, state zipcode Possible keys: Social Security number, street address, city, state, area code, phone number Need str
2、eet address, city, state to uniquely determine location、 A person could have multiple addresses、 The same is true for phones、 These days, a person could have a landline and a cellular phone Exercise 3、1、2 Answers for this exercise may vary because of different interpretations Some possible FDs: ID x
3、-position, y-position, z-position ID x-velocity, y-velocity, z-velocity x-position, y-position, z-position ID Possible keys: ID x-position, y-position, z-position The reason why the positions would be a key is no two molecules can occupy the same point 、Exercise 3、1、3a The superkeys are any subset t
4、hat contains A1、 Thus, there are 2(n-1) such subsets, since each of the n-1 attributes A2 through An may independently be chosen in or out、Exercise 3、1、3b The superkeys are any subset that contains A1 or A2、 There are 2(n-1) such subsets when considering A1 and the n-1 attributes A2 through An、 Ther
5、e are 2(n-2) such subsets when considering A2 and the n-2 attributes A3 through An、 We do not count A1 in these subsets because they are already counted in the first group of subsets、 The total number of subsets is 2(n-1) + 2(n-2)、Exercise 3、1、3c The superkeys are any subset that contains A1,A2 or A
6、3,A4、 There are 2(n-2) such subsets when considering A1,A2 and the n-2 attributes A3 through An、 There are 2(n-2) 2(n-4) such subsets when considering A3,A4 and attributes A5 through An along with the individual attributes A1 and A2、 We get the 2(n-4) term because we have to discard the subsets that
7、 contain the key A1,A2 to avoid double counting、 The total number of subsets is 2(n-2) + 2(n-2) 2(n-4)、Exercise 3、1、3d The superkeys are any subset that contains A1,A2 or A1,A3、 There are 2(n-2) such subsets when considering A1,A2 and the n-2 attributes A3 through An、 There are 2(n-3) such subsets w
8、hen considering A1,A3 and the n-3 attributes A4 through An We do not count A2 in these subsets because they are already counted in the first group of subsets、 The total number of subsets is 2(n-2) + 2(n-3)、Exercise 3、2、1a 精品资料 - - - 欢迎下载 - - - - - - - - - - - 欢迎下载 名师归纳 - - - - - - - - - -第 1 页，共 23
9、页 - - - - - - - - - - 数据库系统基础教程第三章答案We could try inference rules to deduce new dependencies until we are satisfied we have them all、 A more systematic way is to consider the closures of all 15 nonempty sets of attributes、For the single attributes we have A+ = A, B+ = B, C+ = ACD, and D+ = AD 、 Thus,
10、 the only new dependency we get with a single attribute on the left is CA、Now consider pairs of attributes: AB+ = ABCD, so we get new dependency ABD、 AC+ = ACD, and ACD is nontrivial、AD+ = AD, so nothing new、 BC+ = ABCD, so we get BCA, and BCD、 BD+ = ABCD, giving us BDA and BDC、 CD+ = ACD, giving CD
11、A、For the triples of attributes, ACD+ = ACD, but the closures of the other sets are each ABCD 、 Thus, we get new dependencies ABCD, ABDC, and BCDA、Since ABCD+ = ABCD, we get no new dependencies、The collection of 11 new dependencies mentioned above are: CA, ABD, ACD, BCA, BCD, BDA, BDC, CDA, ABCD, AB
12、DC, and BCDA、Exercise 3、2、1b From the analysis of closures above, we find that AB, BC, and BD are keys、 All other sets either do not have ABCD as the closure or contain one of these three sets、Exercise 3、2、1c The superkeys are all those that contain one of those three keys、 That is, a superkey that
13、is not a key must contain B and more than one of A, C, and D、 Thus, the (proper) superkeys are ABC, ABD, BCD, and ABCD、Exercise 3、2、2a i) For the single attributes we have A+ = ABCD, B+ = BCD, C+ = C, and D+ = D 、 Thus, the new dependencies are AC and AD、Now consider pairs of attributes: AB+ = ABCD,
14、 AC+ = ABCD, AD+ = ABCD, BC+ = BCD, BD+ = BCD, CD+ = CD、 Thus the new dependencies are ABC, ABD, ACB, ACD, ADB, ADC, BCD and BDC、For the triples of attributes, BCD+ = BCD, but the closures of the other sets are each ABCD 、 Thus, we get new dependencies ABCD, ABDC, and ACDB、Since ABCD+ = ABCD, we get
15、 no new dependencies、The collection of 13 new dependencies mentioned above are: AC, AD, ABC, ABD, ACB, ACD, ADB, ADC, BCD, BDC, ABCD, ABDC and ACD B、ii) For the single attributes we have A+ = A, B+ = B, C+ = C, and D+ = D 、 Thus, there are no new dependencies、Now consider pairs of attributes: AB+ =
16、ABCD, AC+ = AC, AD+ = ABCD, BC+ = ABCD, BD+ = BD, CD+ = ABCD、 Thus the new dependencies are ABD, ADC, BCA and CDB、For the triples of attributes, all the closures of the sets are each ABCD、 Thus, we get new dependencies ABCD, ABDC, ACDB and BCDA、Since ABCD+ = ABCD, we get no new dependencies、The coll
17、ection of 8 new dependencies mentioned above are: ABD, ADC, BCA, CDB, ABCD, ABDC, ACDB and BCDA、iii) For the single attributes we have A+ = ABCD, B+ = ABCD, C+ = ABCD, and D+ = ABCD 、 Thus, the new dependencies are AC, AD, BD, BA, CA, CB, DB and DC、精品资料 - - - 欢迎下载 - - - - - - - - - - - 欢迎下载 名师归纳 - -
18、 - - - - - - - -第 2 页，共 23 页 - - - - - - - - - - 数据库系统基础教程第三章答案Since all the single attributes closures are ABCD, any superset of the single attributes will also lead to a closure of ABCD、 Knowing this, we can enumerate the rest of the new dependencies、The collection of 24 new dependencies mentioned
19、 above are: AC, AD, BD, BA, CA, CB, DB, DC, ABC, ABD, ACB, ACD, ADB, ADC, BCA, BCD, BDA, BDC, CDA, CDB, ABCD, ABDC, ACDB and BCDA、Exercise 3、2、2b i) From the analysis of closures in 3、2、2a(i), we find that the only key is A、 All other sets either do not have ABCD as the closure or contain A、ii) From
20、 the analysis of closures 3、 2、2a(ii), we find that AB, AD, BC, and CD are keys 、 All other sets either do not have ABCD as the closure or contain one of these four sets 、iii) From the analysis of closures 3、2、2a(iii), we find that A, B, C and D are keys、All other sets either do not have ABCD as the
21、 closure or contain one of these four sets、Exercise 3、2、2c i) The superkeys are all those sets that contain one of the keys in 3、2、2b(i)、 The superkeys are AB, AC, AD, ABC, ABD, ACD, BCD and ABCD、ii) The superkeys are all those sets that contain one of the keys in 3、 2、2b(ii)、 The superkeys are ABC,
22、 ABD, ACD, BCD and ABCD、iii) The superkeys are all those sets that contain one of the keys in 3、2、2b(iii)、The superkeys are AB, AC, AD, BC, BD, CD, ABC, ABD, ACD, BCD and ABCD、Exercise 3、2、3a Since A1A2AnC contains A1A2An, then the closure of A1A2AnC contains B、 Thus it follows that A1A2AnCB、Exercis
23、e 3、2、3b From 3 、2、 3a, we know that A1A2AnCB、 Using the concept of trivial dependencies, we can show that A1A2AnCC、 Thus A1A2AnCBC、Exercise 3、2、3c From A1A2AnE1E2Ej, we know that the closure contains B1B2Bm because of the FD A1A2AnB1B2Bm、 The B1B2Bm and the E1E2Ej combine to form the C1C2Ck、 Thus t
24、he closure of A1A2AnE1E2Ej contains D as well、 Thus, A1A2AnE1E2EjD、Exercise 3、2、3d From A1A2AnC1C2Ck, we know that the closure contains B1B2Bm because of the FD A1A2AnB1B2Bm、 The C1C2Ck also tell us that the closure of A1A2AnC1C2Ck contains D1D2Dj、Thus, A1A2AnC1C2CkB1B2BkD1D2Dj、Exercise 3、2、4a If at
25、tribute A represented Social Security Number and B represented a persons name, then we would assume AB but BA would not be valid because there may be many people with the same name and different Social Security Numbers、Exercise 3、2、4b Let attribute A represent Social Security Number, B represent gen
26、der and C represent name 、 Surely Social Security Number and gender can uniquely identify a persons name (i 、 e、 ABC)、 A Social Security Number can also uniquely identify a person s name (i 、 e、 AC)、 However, gender does not uniquely determine a name (i、e、 BC is not valid)、Exercise 3、2、4c 精品资料 - - -
27、 欢迎下载 - - - - - - - - - - - 欢迎下载 名师归纳 - - - - - - - - - -第 3 页，共 23 页 - - - - - - - - - - 数据库系统基础教程第三章答案Let attribute A represent latitude and B represent longitude、 Together, both attributes can uniquely determine C, a point on the world map (i、e、 ABC)、 However, neither A nor B can uniquely identif
28、y a point (i、e、 AC and BC are not valid)、Exercise 3、2、5 Given a relation with attributes A1A2An, we are told that there are no functional dependencies of the form B1B2 Bn-1C where B1B2 Bn-1 is n-1 of the attributes from A1A2Anand C is the remaining attribute from A1A2An、 In this case, the set B1B2Bn
29、-1 and any subset do not functionally determine C、 Thus the only functional dependencies that we can make are ones where C is on both the left and right hand sides、 All of these functional dependencies would be trivial and thus the relation has no nontrivial FDs、Exercise 3、2、6 Let s prove this by us
30、ing the contrapositive、 We wish to show that if X+ is not a subset of Y+, then it must be that X is not a subset of Y、If X+ is not a subset of Y+, there must be attributes A1A2 An in X+ that are not in Y+、 If any of these attributes were originally in X, then we are done because Y does not contain a
31、ny of the A1A2 An、 However, if the A1A2An were added by the closure, then we must examine the case further、 Assume that there was some FD C1C2CmA1A2Aj where A1A2Aj is some subset of A1A2An、 It must be then that C1C2Cm or some subset of C1C2Cm is in X、However, the attributes C1C2Cm cannot be in Y bec
32、ause we assumed that attributes A1A2Anare only in X+ and are not in Y+、 Thus, X is not a subset of Y、By proving the contrapositive, we have also proved if X ? Y, then X+? Y+、Exercise 3、2、7 The algorithm to find X+ is outlined on pg、 76 、 Using that algorithm, we can prove that (X+)+ = X+、 We will do
33、 this by using a proof by contradiction、Suppose that (X+)+ X+、 Then for (X+)+, it must be that some FD allowed additional attributes to be added to the original set X+、 For example, X+ A where A is some attribute not in X+、 However, if this were the case, then X+ would not be the closure of X、 The c
34、losure of X would have to include A as well、 This contradicts the fact that we were given the closure of X, X+、 Therefore, it must be that (X+)+ = X+ or else X+ is not the closure of X、Exercise 3、2、8a If all sets of attributes are closed, then there cannot be any nontrivial functional dependencies 、
35、 Suppose A1A2、 AnB is a nontrivial dependency、 Then A1A2、 An+contains B and thus A1A2、 An is not closed、Exercise 3、2、8b If the only closed sets are ? and A,B,C,D, then the following FDs hold: AB AC AD BA BC BD CA CB CD DA DB DC ABC ABD ACB ACD ADB ADC BCA BCD BDA BDC CDA CDB ABC D 精品资料 - - - 欢迎下载 -
36、- - - - - - - - - - 欢迎下载 名师归纳 - - - - - - - - - -第 4 页，共 23 页 - - - - - - - - - - 数据库系统基础教程第三章答案ABD C ACD B BCD A Exercise 3、2、8c If the only closed sets are ?, A,B and A,B,C,D, then the following FDs hold: AB BA CA CB CD DA DB DC ACB ACD ADB ADC BCA BCD BDA BDC CDA CDB ABC D ABD C ACD B BCD A Exerc
37、ise 3、2、9 We can think of this problem as a situation where the attributes A,B,C represent cities and the functional dependencies represent one way paths between the cities、 The minimal bases are the minimal number of pathways that are needed to connect the cities、 We do not want to create another r
38、oadway if the two cities are already connected、The systematic way to do this would be to check all possible sets of the pathways、However, we can simplify the situation by noting that it takes more than two pathways to visit the two other cities and come back、 Also, if we find a set of pathways that
39、is minimal, adding additional pathways will not create another minimal set、The two sets of minimal bases that were given in example 3、11 are: AB, BC, CA AB, BA, BC, CB The additional sets of minimal bases are: CB, BA, AC AB, AC, BA, CA AC, BC, CA, CB Exercise 3、2、10a We need to compute the closures
40、of all subsets of ABC, although there is no need to think about the empty set or the set of all three attributes、 Here are the calculations for the remaining six sets: A+=A B+=B C+=ACE AB+=ABCDE AC+=ACE BC+=ABCDE We ignore D and E, so a basis for the resulting functional dependencies for ABC is: CA
41、and ABC、 Note that BC-A is true, but follows logically from C-A, and therefore may be omitted from our list、精品资料 - - - 欢迎下载 - - - - - - - - - - - 欢迎下载 名师归纳 - - - - - - - - - -第 5 页，共 23 页 - - - - - - - - - - 数据库系统基础教程第三章答案Exercise 3、2、10b We need to compute the closures of all subsets of ABC, althou
42、gh there is no need to think about the empty set or the set of all three attributes、 Here are the calculations for the remaining six sets: A+=AD B+=B C+=C AB+=ABDE AC+=ABCDE BC+=BC We ignore D and E, so a basis for the resulting functional dependencies for ABC is: ACB、Exercise 3、2、10c We need to com
43、pute the closures of all subsets of ABC, although there is no need to think about the empty set or the set of all three attributes、 Here are the calculations for the remaining six sets: A+=A B+=B C+=C AB+=ABD AC+=ABCDE BC+=ABCDE We ignore D and E, so a basis for the resulting functional dependencies
44、 for ABC is: ACB and BCA、Exercise 3、2、10d We need to compute the closures of all subsets of ABC, although there is no need to think about the empty set or the set of all three attributes、 Here are the calculations for the remaining six sets: A+=ABCDE B+=ABCDE C+=ABCDE AB+=ABCDE AC+=ABCDE BC+=ABCDE W
45、e ignore D and E, so a basis for the resulting functional dependencies for ABC is: AB, BC and CA、Exercise 3、2、11 For step one of Algorithm 3、7, suppose we have the FD ABCDE 、 We want to use Armstrong s Axioms to show that ABCD and ABCE follow、 Surely the functional dependencies DED and DEE hold beca
46、use they are trivial and follow the reflexivity property、 Using the transitivity rule, we can derive the FD ABCD from the FDs ABCDE and DED、 Likewise, we can do the same for ABCDE and DEE and derive the FD ABCE、For steps two through four of Algorithm 3、7, suppose we have the initial set of attribute
47、s of the closure as ABC、 Suppose also that we have FDs CD and DE、According to Algorithm 3、7, the closure should become ABCDE、 Taking the FD CD and augmenting both sides with attributes AB we get the FD ABCABD 、 We can use the splitting method in step one to get the FD ABCD、 Since D is not in the clo
48、sure, we can 精品资料 - - - 欢迎下载 - - - - - - - - - - - 欢迎下载 名师归纳 - - - - - - - - - -第 6 页，共 23 页 - - - - - - - - - - 数据库系统基础教程第三章答案add attribute D、 Taking the FD DE and augmenting both sides with attributes ABC we get the FD ABCDABCDE 、 Using again the splitting method in step one we get the FD ABCDE、Si
49、nce E is not in the closure, we can add attribute E、Given a set of FDs, we can prove that a FD F follows by taking the closure of the left side of FD F、 The steps to compute the closure in Algorithm 3、7 can be mimicked by Armstrong s axioms and thus we can prove F from the given set of FDs using Arm
50、strongs axioms 、Exercise 3、3、1a In the solution to Exercise 3、2、1 we found that there are 14 nontrivial dependencies, including the three given ones and eleven derived dependencies、 They are: CA, CD, DA, ABD, AB C, ACD, BCA, BCD, BDA, BDC, CDA, ABCD, ABDC, and BCD A、We also learned that the three ke