Four elements are: b, abc, a^2bc^2, a^3bc^3
7) Define a set U of strings of a, b, and c recursively as follows: B....
discreet math 7) Define a set U of strings of a, b, and c recursively as follows:- B. bEU R. If XeU, then axc EU. List four elements of U and apply structural induction to show that every element in U is in the form of a"bc", where n is a non-negative number
discrete math. Structural Induction: Please write and explain clearly. Thank you. Let S be the set of binary strings defined recursively as follows: Basis step: 0ES Recursive step: If r ES then 1rl E S and 0x0ES (I#x and y are binary strings then ry is the concatenation of and y. For instance, if 011 and y 101, then ry 011101.) (a) List the elements of S produced by te first 2 applications of the recursive definition. Find So, Si...
Suppose the language L ? {a, b}? is defined recursively as follows: ? L; for every x ? L, both ax and axb are elements of L. Show that L = L0 , where L0 = {aibj | i ? j }. To show that L ? L 0 you can use structural induction, based on the recursive definition of L. In the other direction, use strong induction on the length of a string in L0. 1.60. Suppose the language...
4. Suppose S is the set of numbers recursively defined by: lE S Use structural induction to prove that all members of S are integers of the form of 2a3 for some non-negative integers a, b.
QUESTION 10 The equality relationon any set S is: A total ordering and a function with an inverse. An equivalence relation and also function with an inverse. A function with an inverse, and an equivalence relation with as single equivalence class equal to S An equivalence relation and also a total ordering QUESTION 11 A binary operation on a set S, takes any two elements a,b E S and produces another element c e S. Examples of binary operations include...
2. 9 marks] Strings. Consider the following definitions on strings Let U be the set of all strings Let s be a string. The length or size of a string, denoted Is, is the number of characters in s Let s be a string, and i e N such that 0 < ί < sl. We write s[i] to represent the character of s at index i, where indexing starts at 0 (so s 0] is the first character, and...
2. 9 marks] Strings. Consider the following definitions on strings Let U be the set of all strings. Let s be a string. The length or size of a string, denoted Is, is the number of characters in s Let s be a string, and ie N such that 0 Si< Is. We write si] to represent the character of s at index i, where indexing starts at 0 (so s(0 is the first character, and s|s -1 is the...
4. (7 pts) Let A be an array of 5 integers, whose contents are as follows: 3, 2, 1, 5, 4. We will apply insertion sort to sort this array. Show all of the element-wise comparisons made by the algorithm in the correct order. Here an element-wise comparison means the comparison of one element of the array with another element of the array or the key set in a particular step of the algorithm. Since the algorithm may move the...
Write a regular expression that captures the set of strings composed of 'a', 'b', and 'c', where any string uses at most two of the three letters (for example, "abbab" is a valid string, or "bccbb", or "ccacaa", but not "abccba": strings that contain only one of the three letters are also fine). Give a non-deterministic finite automaton that captures the regular expression from Using the construction described in class, give a deterministic version of the automaton. Repeat the previous...
c We define the operation * on subsets of a universal set U as follows. For any two sets A and B: A*B:= AUB Answer the following questions using the Laws of Set Operations and any derived results given in lec- tures) to justify your answer: (a) What is (A + B) * (A*B)? (b) Express A using only A, * and parentheses (if necessary). (c) Express using only A, * and parentheses (if necessary). (d) Express A | B...