please answer question 25

T. er of re as ins E4. sed ice, ular '1') 1.3 Sequences 19 Example 18 Let A = (0,1). Find regular sets corresponding to the three regular expressions in Example 16. Solution (a) The set corresponding to O" (O v 1) consists of all sequences of O's and 1's. Thus, the set is A (6) The expression 00" (OV 1)*1 corresponds to the set of all sequences of O's and I's that begin with at least one and end with at least one 1. (e) The expression (01)(OI V 1") corresponds to the set of all sequences of O's and 1's that either repeat the string 01 a total of n > 1 times, or begin with a total of no repetitions of 01 and end with some number k> 0 of 1's. This set includes, for example, the strings 1111, 01,010101, 0101010111111, and 011. 1.3 Exercises In Exercises through 4, give the set corresponding to the se- 23. Let A = (x x is a real number and 0 4), D = x 1 x E R and x + 3x + 2 * 0), and 5. Give three different sequences that have (x, y, z) as a cor- E = {(x, y, z) x E Z. YER*.2 Z*). Identify cach responding set set as finite, countable, or uncountable, 6. Give three different sequences that have (1,2,3,... asa 25. Let A = {ab, be, ba). In each part, tell whether the string corresponding set belongs to A In Exercises through 14, write out the first four terms (begin (a) ababab (b) abc (c) abba with n 1) of the sequence whose general term is given (d) abbchaba (e) bcabbab ( abbbcba 7. , - 26. Let U = (arithmetic, algebra, calculus, geometry, trig- 8. b.-312 + 2n-6 onometry, analysis, topology, statistics), B = (analysis, topology, calculus), C = {algebra, calculus, trigonom- 9. 8. -1.2-3.... etry, analysis), D = (arithmetic, algebra, calculus, topology, statistics), E = {algebra, calculus, geometry, 10. h. trigonometry, analysis). In cach of the following, repre- 11.2.5.6.-1+1.5 sent the given set by an array of zeros and ones. 12. d.-3.d. = -24.1 (a) BUC (b) CND 13.40.41-2 (e) B( DE) (d) FUE 14. = 4= (e) CN(BUE) 27. Let U - 10.0.6.8.,km.), B = (6) C = (4.8. In Exercises 15 through 20, write a formula for the nth term of m, n, and D = d.k.n). the sequence. Identify your formula as recursive or explicit (a) What is /o(b)? (b) What is fele)? 15. 1.3.5.7.... 16.0.3, 8, 15, 24, 35,... (c) Find the sequences of length 8 that correspond to fa 17.1.-1.1.-1.1.-1.... 18. 0.2,0,2,0,2,... fc. and o 19. 1.4.7.10, 13, 16 20. 1... (d) Represent BUC, CU D, and CD by arrays of zeros and ones. 21. Write an explicit formula for the sequence 2,5,8, 11. 28. Complete the proof that fros = A + fa-2fafa 22. Write a recursive formula for the sequence 2.5.7.12. [Theorem 1(c)]. Suppose x E A and B. Then fACX) = and f(x)f(x) so f1(x) + f(x) - 28.(x)/xx) = sub- fno ular e in N of IN fA cor nents es be of all ssion ssion repeat bcbc. 14,17, 19.31