PLEASE USE INDUCTION, NO PROGRAMS ALLOWED
Take M={P=anxn+…+a0∈Z[x]/an>0}, with the usual addition and multiplication on polynomials. Interprete S(P) as P(X)+1.
Then M is a model of Robinson arithmetic, but there are long strings of "not-even" numbers, such as X,X+1,X+2,…. So in this model, it is false that if n is not even then n+1 is even.
If you define "n is odd" as "∃k/n=k+k+1", then it is false that every number is even or odd.
However, it is still true in M that addition is associative and commutative, and so if n is odd then n+1 is even, and if n is even, then n+1 is odd. (you will need a stranger model for this to fail)
If you want a model in which a2−2b2=0 has a solution, you can pick M={P∈Z[X,Y]/(X2−2Y2)/limy→∞P(√2y,y)=+∞ or P∈N}
PLEASE USE INDUCTION, NO PROGRAMS ALLOWED Consider the following program, where a and n are positive...
Problem Description proving program correctness Consider the following program specification: Input: An integer n > 0 and an array A[0..(n - 1)] of n integers. Output: The smallest index s such that A[s] is the largest value in A[0..(n - 1)]. For example, if n = 9 and A = [ 4, 8, 1, 3, 8, 5, 4, 7, 2 ] (so A[0] = 4, A[1] = 8, etc.), then the program would return 1, since the largest value in...
Consider a discrete-time system that is linear (but not necessarily time-invariant), and where: - if the input x[n] is even, then the output is y[n]= x[n-1] - if the input x[n] is odd, then the output y[n]= x [n=1] Find the ouput of this system if the inpus is (a) δ [n] (b) u[n]. (Do not use la place transform) Hint: if a signal is neither even nor odd, then you can write it as a sum of an even...
Use a java program that does the following:
. (10 points) Write a program as follows a. Prompt the user to input two positive integers: nl and n2 (nl should be less than n2) b. If the user enters the negative number(s), convert it/them to positive number(s) c. If nl is greater than n2, swap them. d. Use a while loop to output all the even numbers between nl and n2 e. Use a while loop to output the sum...
Please
answer all of the questions.
6. Consider three systems with the following input-output relationships: { 0, odd System 1: yn 피[핑], n even System 2: y[nx[n] - 10xr[n + 2] + 3xr[n - 1 System 3: yn x[3n] The interconnection diagram is at follows: y System 3 System System 2 Find the input-output relationship of the interconnected system. State the properties of the system (linear, stable, time invariant, memoryless, and causal)
6. Consider three systems with the following input-output...
Write a Java program to prompt for inputting an integer N, then enter N integers (a loop is needed), print the numbers user entered, and the amount of even and odd numbers (zero is even number). (1) Prompt for the user to input an integer and output the integer. (1 pts) Enter an integer: You entered: 5 (2) Prompt for the user to input N integers, output the numbers entered and the amount of even and odd numbers (9 pts)...
Consider three systems with the following input-output
relationships
6. Consider three systems with the following input-output relationships: { 4 0, odd System 1: y[n n even r[n] 10ar(n 2]3r[n - 1 System 2: yn + + System 3: yn x[3n] The interconnection diagram is at follows: System 1 System 2 System 3 Find the input-output relationship of the interconnected system. State the properties of the system (linear, stable, time invariant, memoryless, and causal).
6. Consider three systems with the following...
Tems.] Use the second principle of induction to prove that every positive integer n has a factorization of the form 2m, where m is odd. (Hint: For n > 1, n is either odd or is divisible by 2.)
Complete the Python program below that performs the following operations. First prompt the user to input two integers, n and m. If n<m, then print the odd positive integers that are less than m (in order, on a single line, separated by spaces). If man, then print the even positive integers that are less than n (in order, on a single line, separated by spaces). If neem, then print nothing. For instance, if the user enters 5 followed by 10,...
Student ID: 123
Write a C+ program with the following specifications: a. Define a C++ function (name it function_StudentlD where StudentID is your actual student ID number) that has one integer input (N) and one double input (x) and returns a double output S, where N S = n 0 and X2 is given by 0 xeVn n 0,1 Хл —{2. nx 2 n 2 2 m2 x2 3 (Note: in the actual quiz, do not expect a always, practice...
(a) Prove the following loop invariant by induction on the
number of loop iterations: Loop Invariant: After the kth iteration
of the for loop, total = a1 + a2 + · · · + ak and L contains
all elements from a1 , a2 , . . . ,
ak that are greater than the sum of all previous terms of the
sequence.
(b) Use the loop invariant to prove that the algorithm is
correct, i.e., that it returns a...