Problem 1E: t the ordered pairs in the relationRfromA=0,1,2,3,4)toB=0,1,2,3), where(a,b)Rif and only if a)a=b.... Problem 2E: a) List all the ordered pairs in the relation R = {(a, b) |adividesb} on the set {1, 2, 3,4,5, 6}.... Problem 3E: each of these relations on the set {1, 2, 3, 4}, decide whether it is reflexive, whether it is... Problem 4E: ermine whether the relationRon the set of all people is reflexive, symmetric, antisymmetric, and/or... Problem 5E: ermine whether the relationRon the set of all Web pages is reflexive, symmetric, antisymmetric,... Problem 6E: ermine whether the relationRon the set of all real numbers is reflexive, symmetric, antisymmetric,... Problem 7E: ermine whether the relationRon the set of all integers is reflexive, symmetric, anti-symmetric,... Problem 8E: w that the relationR=Oon a nonempty set S is symmetric and transitive, but not reflexive. Problem 9E: Show that the relationR=on the empty setS=is reflexive, symmetric, and transitive. Problem 10E: e an example of a relation on a set that is a) both symmetric and antisymmetric. b) neither... Problem 11E: Which relations in Exercise 3 are irreflexive? Problem 12E: Which relations in Exercise 4 are irreflexive? Problem 13E: Which relations in Exercise 5 are irreflexive? Problem 14E: Which relations in Exercise 6 are irreflexive? Problem 15E: Can a relation on a set be neither reflexive nor irreflexive? Problem 16E: Use quantifiers to express what it means for a relation to be irreflexive. Problem 17E: Give an example of an irreflexive relation on the set of all people. A relationRis called asymmetric... Problem 18E: Which relations in Exercise 3 are asymmetric? Problem 19E: Which relations in Exercise 4 are asymmetric? Problem 20E: Which relations in Exercise 5 are asymmetric? Problem 21E: Which relations in Exercise 6 are asymmetric? Problem 22E: Must an asymmetric relation also be antisymmetric? Must an antisymmetric relation be asymmetric?... Problem 23E: Use quantifiers to express what it means for relation to be asymmetric. Problem 24E: Give an example of an asymmetric relation on the set of all people. Problem 25E: many different relations are there from a set withmelements to a set withnelements? •.. LetRbe a... Problem 26E: Rbe the relationR={(a,b)ab}on the set of integers. Find a)R. b)R. Problem 27E: Rbe the relationR={(a,b) |adividesb} on the set of positive integers. Find a)R b)R Problem 28E: Let R be the relation on the set of all states in the United States consisting of pairs (a, b) where... Problem 29E: pose that the functionffromAtoBis a one-to-one correspondence. LetRbe the relation that equals the... Problem 30E: R1= {(1, 2), (2, 3), (3, 4)} andR2= {(1, 1), (1, 2), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3,3),... Problem 31E: Abe the set of students at your school andBthe set of books in the school library. LetR1andR2be the... Problem 32E: Rbe the relation {(1, 2), (1, 3), (2, 3), (2,4), (3, 1)}, and let S be the relation {(2, 1), (3, 1),... Problem 33E: 33.LetRbe the relation on the set of people consisting of pairs (a,b),whereais a parent ofb.LetSbe... Problem 34E: rcises 34-38 deal with these relations on the set of real numbers: R1={(a,b)R2ab},the greater than... Problem 35E: rcises 34-38 deal with these relations on the set of real numbers: R1={(a,b)R2ab},the greater than... Problem 36E: rcises 34-38 deal with these relations on the set of real numbers: R1={(a,b)R2ab},the greater than... Problem 37E: rcises 34-38 deal with these relations on the set of real numbers: R1={(a,b)R2ab},the greater than... Problem 38E: rcises 34-38 deal with these relations on the set of real numbers: R1={(a,b)R2ab},the greater than... Problem 39E: d the relationsS2fori= 1, 2, 3,4, , 6i’here S1={(a,b)Z2ab},the greater than relation,... Problem 40E: Rbe the parent relation on the set of all people (seeExample 21).When is an ordered pair in the... Problem 41E: Rbe the relation on the set of people with doctorates such that(a,b)Rif and only ifawas the thesis... Problem 42E: R1andR2be the divides” and ‘is a multiple of relations on the set of all positive integers,... Problem 43E: R1andR2be the “congruent modulo 3” and the “congruent modulo 4” relations, respectively, on the set... Problem 44E: List the 16 different relations on the set {0,1}. Problem 45E: How many of the 16 different relations on {0,1} contain the pair (0,1)? Problem 46E: ch of the 16 relations on {o, 1}, which you listed in Exercise 44, are a) reflexive? b) irreflexive?... Problem 47E: a) How many relations are there on the set {a,b,c,d}? b) How many relations are there on the... Problem 48E: S be a set withnelements and letaandbbe distinct elements ofS. How many relationsRare there onSsuch... Problem 49E: How many relations are there on a set withnelements that are a) symmetric? b) antisymmetric? c)... Problem 50E: How many transitive relations are there on a set withnelements if a)n=1? b)n=2? c)n=3? Problem 51E: d the error in the “proof” of the following “theorem.” “Theorem”:LetRbe a relation on a setAthat is... Problem 52E: pose thatRandSare reflexive relations on a setA. Prove or disprove each of these statements. a)RSis... Problem 53E: w that the relationRon a setAis symmetric if and only ifR=R1, whereR1R is the inverse relation. Problem 54E: w that the relationRon a setAis antisymmetric if and only ifRR1is a subset of the diagonal... Problem 55E: w that the relationRon a setAis reflexive if and only if the inverse relationR1is reflexive. Problem 56E: w that the relationRon a setAis reflexive if and only if the complementary relationRis irreflexive. Problem 57E: Rbe a relation that is reflexive and transitive. Prove thatRn=Rfor all positive integersn. Problem 58E: Rbe the relation on the set {1, 2, 3,4 , 5} containing the ordered pairs (1, 1), (1, 2), (1, 3), (2,... Problem 59E: Rbe a reflexive relation on a setA. Show thatRnis reflexive for all positive integersn. Problem 60E Problem 61E: Suppose that the relationRis irreflexive. IsR2necessarily irreflexive? Give a reason for your... Problem 62E: ive a big-O estimate for the number of integer comparisons needed to count all transitive relations... format_list_bulleted