suppose that x and y are bit strings of length n and m is the number of positions where both x and y have 1s. show that w(x y) = w(x) w(y)− 2m.

Answers

Answer 1

A bit string of length refers to a sequence of bits, each of which can be either 0 or 1, that is a fixed length. The total number of possible bit strings of length n is 2^n.

To start, let's define what the function w() represents. The function w(x) returns the Hamming weight of bit string x, which is the number of positions in the string where the bit is 1.

Now, we need to prove that w(x y) = w(x) w(y)− 2m, where x and y are bit strings of length n and m is the number of positions where both x and y have 1s.

First, let's consider the case where x and y have no 1s in common. In this case, m=0 and w(x y) will be equal to the Hamming weight of the concatenated string x y, which is the sum of the Hamming weights of x and y.

This is because there are no positions in which both x and y have 1s, so the number of 1s in x y will be equal to the number of 1s in x plus the number of 1s in y.

Now, let's consider the case where x and y have some 1s in common. In this case, we need to subtract the number of overlapping 1s from the sum of the Hamming weights of x and y, since these overlapping 1s will be double-counted in the concatenated string x y. The number of overlapping 1s is equal to 2m, since each position where both x and y have a 1 will be counted twice in the concatenated string.

Therefore, we can express w(x y) as w(x) + w(y) - 2m, which is equivalent to w(x) w(y) - 2m. This completes the proof.

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Related Questions

This question involves the creation and use of a spinner to generate random numbers in a game. a gamespinner object represents a spinner with a given number of sectors, all equal in size. the gamespinner class supports the following behaviors.

creating a new spinner with a specified number of sectors
spinning a spinner and reporting the result
reporting the length of the current run, the number of consecutive spins that are the same as the most recent spin

write the complete gamespinner class. your implementation must meet all specifications and conform to the example.

Answers

Using the knowledge in computational language in JAVA it is possible to write a code that generate random numbers in a game

Writting the code in JAVA:

public class GameSpinner {

int spinner;

int num;

int current;

public GameSpinner(int spinner) {

super();

this.spinner = spinner;

this.current = 0;

this.num = 0;

}

public int spin() {

int curr = 1 + (int) (Math.random() * spinner);

if (this.current == 0 || this.num == 0) {

this.current = 1;

this.num = curr;

return curr;

}

if (this.num == curr) {

this.current = this.current + 1;

return curr;

} else {

this.num = curr;

this.current = 1;

return curr;

}

public int currentRun() {

return this.current;

}

GameSpinnerTest.java

public class GameSpinnerTest {

public static void main(String[] args) {

// TODO Auto-generated method stub

GameSpinner g = new GameSpinner(4);