Let A= {1,2,3,4}, B={5,6,7} C={8,9,10,11} and f: A-B g: B C h: functions are 1-1 a) Form the function f, g, h Apply the definitions showed in class and find the following b) gof c) fog
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- Computer Science In function f(x) ->y the domain of x is continuous in the range 1-10how many equivalence classes are there in this case?) Let O be the set of odd numbers and O’ = {1, 5, 9, 13, 17, ...} be its subset. Definethe bijections, f and g as:f : O 6 O’, f(d) = 2d - 1, d 0 O.g : ø 6 O, g(n) = 2n + 1, n 0 ø.Using only the concept of function composition, can there be a bijective map from øto O’? If so, compute it. If not, explain in details why not...................................................................................................................................... [2+8]b) A Sesotho word cannot begin with of the following letters of alphabet: D, G, V, W,X, Y and Z.We define the relation: A Sesotho word x is related to another Sesotho word y if xbegins with the same letter as y.Determine whether or not this is an equivalence relation.If it is an equivalence relation then1. Compute C(sekatana)2. How many equivalence classes are there in all, and why?3. What is the partition of the English words under this relation?If it is NOT an equivalence relation then explain in details why it is notYou should simply fill in the blanks for part (i), and briefly justify your answers for parts (ii), (iii) and (iv). i. Let S ⊆ ATM. What do the elements of S look like ?e.g. “An element of S is a pair such that ”. ii. Define a subset S of AT M which is infinite and decidable.Note: You need to explicitly say/define here which elements of A are in your set S, not just that such an S exists. iii. T or F: For all languages L and J, if L is a recognizable but undecidable language, and J is an infinite decidable subset of L then L-J is undecidable. Explain your answer. iv. T or F: The complement of AT M also contains an infinite decidable subset. Explain your answer
- Subject - Theory of Computing ALEX is the set of valid algebraic expressions recursively defined by:Rule 1: All polynomials are in ALEXRule 2: If f(x) and g(x) are in ALEX, then so are:i. (f(x))ii. -(f(x))iii. f(x) + g(x)iv. f(x) - g(x)v. f(x)g(x)vi. f(x)/g(x)vii. f(x)g(x)viii. f(g(x))Assuming that the rules seen in class recursively defining polynomials can be used here to prove x+2 and 3x are polynomials, show that (x + 2)3x is in ALEX.2. Let x and y be two words (both different than the empty string) and xy is their concatenation. Show that if x, y and xy are all in PALINDROME, then there is a word z such that x= zn and y=zm for some integers n and m.Prove that the next issue is in the NP class: An integer t and a collection of integers S have been provided to us. Is there a subset of S whose elements add up to t? Please take note: Algorithm and Data Structures IssuePlease answer the following question in depth with full detail. Consider the 8-puzzle that we discussed in class. Suppose we define a new heuristic function h3 which is the average of h1 and h2, and another heuristic function h4 which is the sum of h1 and h2. That is, for every state s ∈ S: h3(s) =h1(s) + h2(s) 2 h4(s) =h1(s) + h2(s) where h1 and h2 are defined as “the number of misplaced tiles”, and “the sum of the distances of the tiles from their goal positions”, respectively. Are h3 and h4 admissible? If admissible, compare their dominance with respect to h1 and h2, if not, provide a counterexample, i.e. a puzzle configuration where dominance does not hold.
- Mathematical Logic First-order or predicate logic. Show that the sum relation, {(m,n,p)|p=m+n}, is not definable on (N; ∙). Hint: Consider an automorphism of (N; ∙) that interchanges two primes. Where N is the set of natural numbers. Please be as clear as possible. Show and explain all the steps. Thank you very much.Consider a robot that is capable of picking objects from a shelf and placing them on the table. Suppose that the robot’s arm works perfectly, and the environment is fully-observable. (i) Define a STRIPS operator that could be used for planning the actions of this robot. (ii) Give an example of a state S of the world at which this STRIPS operator is applicable. (iii) Describe the operation of this STRIPS operator at this state S, to show the next state S0 of the world.Write a java program using OOP concept that takes in pair of from two list from the domain and target Domain A={a, b, c, d} Target B={1, 2, 3, 4} Based on the inputs, determine if the function Injective, Surjective, and Bijective mapping-------------------------------Sample runa 1b 2c 3d 4 The function is InjectiveThe function is SurjectiveThe function is Bijective------------------------------- Sample runa 1b 2c 1d 3 The function is Surjective
- The homomorphism h is defined by h(a) = 01 and h(b) = 10. What is h(%%1%%2%%3%%4)?4. Consider the function second xs = head (tail xs) What is its type? Explain why "second" has the type that it has, i.e., "tail :: [a] ->[a]", hence "xs :: [a]", and "head :: [a] -> a", etc. Why doeschaining them together like in the definition of "second" give us thetype that it has? Note: this will use the fact that if "f" is a function that mapsarguments of type "a" to results of type "b", and "e" is an expressionof type "a", then the application "f e" has type "b", that is: if "f :: a -> b" and "e :: a" then "f e :: b" 5. Consider the function swap (x,y) = (y,x) What is its type? Why can "x" and "y" correspond todifferent types in general?A tile of a monkey puzzle has four monkey halves that can be labelled as north(N), east (E), south (S), and west (W) half. In addition to the shape of the borderrectangle, these halves determine which edges can be placed next one other. There isalso another way to define how the tiles can be placed: Each tile corner (i.e. compassdirections NE, SE, SW, and NW) has a monkey quarter. If we abstract this quarter, forexample, with a letter, only the tiles with the same letter in their touching corners canbe adjacent. illustrates one valid solution for this quarter monkey puzzle.Are the two monkey puzzle representations equivalent in the sense that if we have apile of ‘half monkey’ tiles H, it is possible to define a pile of ‘quarter monkey’ tilesQ that gives exactly the same set of solutions for the puzzle (and vice versa)?