1. Suppose that C is generated from a set B= {a,b} by the binary operation f and the unitoperation g . List all the elements of C, . How many elements can they have C, y C,?Remember that C, is the set of all points x such that some construction sequence oflength n ends with x, and soB = CcC,cC; s..2. Prove that (A, →^4,) it is not a formula

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Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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7.(a) For a set R, addition (+) and multiplication (*) operations classify the algebraic structure (1 (R, +, *). Show the steps of your findings. If R is replaced by R*, what will happen to your structure and why?
Exercise 1.8. Prove that if one chooses n+1 numbers from {1, 2,3, ...,2n}, it is guaranteed that two of the numbers they chose are consecutive. Also, before your proof, write down an example of 4 numbers from {1,2, 3, 4, 5, 6} and locate two of them which are consecutive. Then, repeat for 5 numbers from {1, 2,...,8} and 6 numbers from {1,2, ..., 10}.
Consider z and the binary relation on A given by R=( (x.y)lx+y = 10) For each item, fill the blank by using Y if the binary relation satisfies the property and N if the binary relation does not satisfy the property 1.( ) Reflexive 2( ) Symmetric 3. ( )Antisymmotric 4. ( ) Transitive
4. Consider the set of all strings of a's, b's, c's and d's. (a) Make a list of all of these strings of length zero, one, and two that do not contain the pattern bb. (b) For each integer n > 0, let sn = the number of strings of a's, b's, c's and d's of length n that do not contain the pattern bb. Find so, s1, and s2. (c) Find a recurrence relation for the sequence s0, S1, S2, . ..
Let S be the set of all bit strings (strings of 0's and I's) Select one: O A. f(s) = the string obtained by moving the first bit of s to the end of the string. (For example, f(1001101) = 0011011) O B. f(s) = the string obtained from s by interchanging 0's and I's. (For example, f(11000) = 00111) O c. f(s) the string & with a 1 bit appended at the end. (For example, f(1101) = 11011) O D. f(s) the reversal of s. (For example, f(110) = 011) length at least 2. Which of the following functions f : S → S is not onto S?
Let S be the set of all strings of 0's and 1's of length 4, and let A and B be the following subsets of S. A = {0111, 1011, 0100, 0110} and B = {0010, 1010, 0111, 1110} Find each of the following. (Enter your answers in set-roster notation.) (a) AnB = (b) AUB= (c) A - B = (d) B-A = X X
6) Let A = {4,5,6} and B = {3,6,7}. Define relations U, V, and W from A to B as follows, For all (x, y) E A×B, (x, y) eU x>y (x, y) e V > X-y is an integer. W = {(4,7), (6,3), (5,6)} Indicate whether any of the relations U, V, and W are functions from A to B.
i. Determine whether (~p^(p→q)→~q)is a tautology. ii. Define Cartesian product. iii. List the ordered pairs in the relation R from A = {0, 1, 2, 3, 4} to B = {0, 1, 2, 3}, where (a, b) E R if and only if gcd (a,b) =1 iv. How many relations are there on the set {a,b,c,d} Are the following relations on {0, 1, 2, 3} are equivalence relations or v. partial ordered relation? Explain? {(0, 0), (1, 1), (2, 2), (3, 3)}
Let S be the set [9]{1.2. 9). (a) How many derangements has the set S? (b) How many derangements of S have 9 in the 2nd position and 1 in the last position? (c) List 5 derangements of S having 2 in the 3rd position and 5 in the 2nd position. 101

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