Let S be the set of all strings of O's and 1's, and define D:S - →→→ Z as follows: For every s ES, D (s) = = the number of 1's minus the number of O's in s. Which one of the following statements about D is true? D is not 1-1 but is onto. D is 1-1 and onto. D is not 1-1 and not onto. D is 1-1 but not onto.

Let S be the set of all strings of O's and 1's, and define D:S - →→→ Z as follows: For every s ES, D (s) = = the number of 1's minus the number of O's in s. Which one of the following statements about D is true? D is not 1-1 but is onto. D is 1-1 and onto. D is not 1-1 and not onto. D is 1-1 but not onto.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.1: Sets

Problem 13E: 13. Let Z denote the set of all integers, and let Prove that .

See similar textbooks

Similar questions

13. Let Z denote the set of all integers, and let Prove that .
Assume the statement from Exercise 30 in section 2.1 that for all and in . Use this assumption and mathematical induction to prove that for all positive integers and arbitrary integers .
Label each of the following statements as either true or false. 3. Let where A and B are nonempty. Then for every subset S of A.
Let x and y be integers, and let m and n be positive integers. Use mathematical induction to prove the statements in Exercises 1823. ( The definitions of xn and nx are given before Theorem 2.5 in Section 2.1 ) (m+n)x=mx+nx
12. (See Exercise 10 and 11.) If each is identified with in prove that . (This means that the order relation defined in Exercise 10 coincides in with the original order relation in . We say that the ordering in is an extension of the ordering in .) 11. (See Exercise 10.) According to Definition 5.29, is defined in by if and only if . Show that if and only if . 10. An ordered field is an ordered integral domain that is also a field. In the quotient field of an ordered integral domain define by . Prove that is a set of positive elements for and hence, that is an ordered field. Definition 5.29 Greater than Let be an ordered integral domain with as the set of positive elements. The relation greater than, denoted by is defined on elements and of by if and only if . The symbol is read “greater than.” Similarly, is read “less than.” We define if and only if. As direct consequences of the definition, we have if and only if and if and only if . The three properties of in definition 5.28 translate at once into the following properties of in . If and then . If and then . For each one and only one of the following statements is true: . The other basic properties of are stated in the next theorem. We prove the first two and leave the proofs of the others as exercises.