离散(英)chap4作业答案.doc

Chap4 作业答案P44, 1.613.Yes, the nn zero matrix, which is a diagonal matrix.14.Yes,单位矩阵。15. is the diagonal matrix with i,ith entry 16. is the diagonal matrix with i,ith entry P333, 9.11.yes 2.no 3.no 4.yes 5.no 6.no 7.no 8.mutative, associative16.not commutative, not associative23.28. Since a binary operation on is a function : , there are binary operations can be defined on .29.Only elements can be chosen, and everyone has choices, then there are commutative binary operations can be defined on .P340, 9.22. First, verify whether the operations commutative, Hence, both the operations are associative.The identity of is , hence it is a monoid;The identity of is , hence it is also a monoid.6. Its a commutative semigroup.9. is closed on ; is associative; the identity is , and is commutative, hence it is a commutative monoid.13. is closed on ;,so , hence is associative; Obviously, is commtative;, is the identity;after all, it is a commutative monoid.18. , , then is not associative, it is not a semigroup.20. 26.Proof: Let be the subsemigroup of 30. Let , , is not empty and has the identity;For is associative on , and , therefore, it is associative on , too;For any , , , and is commutative on , thereforehence , is closed on .P357, 9.48. Abelian group: closed: , for ;associative and commutative;the identity is ;the inverse of is .10. Abelian group 封闭，结合，交换，单位元：零阵，逆元：-A18. For any , , ,Since is a group, there exist , and is associative, then,or,or,it follows that is Abelian.或证： 21.Consider the sequence Since G is finite, not all terms of this sequence can be distinct,That is, for some . Then ,and .25. 证 设,即对,有 ,于是 ,从而 ,于是 ,即构成的子群.26. 证 ,非空,结合律显然;若,则,即,封闭；若,即, ,即,综上所述,是的子群.33.:Suppose defined by is a homomorphism. Then Hence : Suppose is Abelian, Then So homomorphism.34. 充分性：设是一