462 ALGEBRAIC SYSTEMS [APP. B
B.41. LetAbe a nonempty set with the operation∗defined bya∗b=a, and assumeAhas more than one element.
(a) IsAa semigroup? (c) DoesAhave an identity element?
(b) IsAcommutative? (d) Which elements, if any, have inverses and what are they?
B.42. LetA={a, b}. (a) Find the number of operations onA. (b) Exhibit one which is neither associative nor com-
mutative.
B.43. For each of the following sets, state which are closed under: (a) multiplication; (b) addition.A={ 0 ,l},B={ 1 , 2 },C={x|x is prime}, D={ 2 , 4 , 8 ,...}={x|x= 2 n}.B.44. LetA={...,−9,−6,−3, 0, 3, 6, 9, ...}, the multiples of 3. IsAclosed under:
(a) addition; (b) multiplication; (c) subtraction; (d) division (except by 0)?B.45. Find a setAof three integers which is closed under: (a) multiplication; (b) addition.
B.46. LetSbe an infinite set. LetAbe the collection of finite subsets ofSand letBbe the collection of infinite subsets
ofS
(a) IsAclosed under: (i) union; (ii) intersection; (iii) complements?
(b) IsBclosed under: (i) union; (ii) intersection; (iii) complements?B.47. LetS=Q×Q, the set of ordered pairs of rational numbers, with the operation∗defined by(a, b)∗(x, y)=(ax, ay+b)(a) Find (3, 4)∗(1, 2) and (−1, 3)∗(5, 2). (c) Find the identity element ofS.
(b) IsSa semigroup? Is it commutative? (d) Which elements, if any, have inverses and what are they?
B.48. LetS=N×N, the set of ordered pairs of positive integers, with the operation∗defined by(a, b)∗(c, d)=(ad+bc, bd)(a) Find (3, 4)∗(1, 5) and (2, 1)∗(4, 7).
(b) Show that∗is associative. (Hence thatSis a semigroup.)
(c) Definef:(S,∗)→(Q,+)byf(a,b)=a/b. Show thatfis a homomorphism.
(d) Find the congruence relation∼inSdetermined by the homomorphismf, that is,x∼yiff(x)=f(y).
(e) DescribeS/∼. DoesS/∼have an identity element? Does it have inverses?
B.49. LetS=N×N. Let∗be the operation onSdefined by(a, b)∗(a′,b′)=(a+a′,b+b′)(a) Find (3, 4)∗(1, 5) and (2, 1)∗(4, 7).
(b) Show that∗is associative. (Hence thatSis a semigroup.)
(c) Definef:(S,∗)→(Z,+)byf(a,b)=a−b. Show thatfis a homomorphism.
(d) Find the congruence relation∼inSdetermined by the homomorphismf.
(e) DescribeS/∼. DoesS/∼have an identity element? Does it have inverses?GROUPS
B.50. ConsiderZ 20 ={ 0 , 1 , 2 ,..., 19 }under addition modulo 20. LetHbe the subgroup generated by 5. (a) Find the
elements and order ofH. (b) Find the cosets ofHinZ 20.
B.51. ConsiderG={ 1 , 5 , 7 , 11 }under multiplication modulo 12. (a) Find the order of each element. (b) IsGcyclic?
(c) Find all subgroups ofG.
B.52. ConsiderG={ 1 , 5 , 7 , 11 , 13 , 17 }under multiplication modulo 18. (a) Construct the multiplication table ofG.
(b) Find 5−^1 ,7−^1 , and 17−^1. (c) Find the order and group generated by: (i) 5; (ii) 13; (d) IsGcyclic?