Please include explanations to each answer. Find the distance between x and y. x
ID: 3795402 • Letter: P
Question
Please include explanations to each answer.
Find the distance between x and y. x = 110110 and y = 000101 3 2 4 6 You receive the following binary message: 1001001. The message includes three check bits. Was there any bit erroneously received and if so which one? a1 = 001 a2 = 010 a3 = 011 a5 = 101 You receive the following binary message: 1001001. The message includes three check bits. What was the correct number (in decimal system) sent? 2 9 5 1 you receive the following binary message: 1001101. The message induces three check bits. Was there any bit erroneously received and if so which one? a1 = 001 a3 = 011 a7 = 111 a6 = 110 you receive the following binary message: 1001101. The message includes three check bits. What was the correct number (in decimal system) sent? 9 4 5 1 Proof that, if n is an integer and n^2 is odd, then n is odd. True False Use direct proof using definitions only or give counter-example. Proof that for all integers, if a is even and b is odd then a^2 - 3b is odd. True FalseExplanation / Answer
Q1
A: Question 1 is answered by ivolving the mechanism Hamming Distance(HD).
HD helps to identify the distance between two binary strings of equal length.
In our case if we consider the distance between X= 110110 and Y = 000101.
The answering process takes the following steps:
Step 1
Primarily, make sure the two strings are of equal length. The Hamming distance can only be calculated between two strings of equal length i.e X: "110110" and Y: "000101"
Step 2
Compare the first bit of one string to the first bit of other string. If they are the same, record a "0" for that bit. If they are different, record a "1" for that bit. In this case, the first bit of both strings is "1," so record a "0" for the first bit.
Step 3
Compare each bit in succession and record either "1" or "0" as appropriate. X: "110110" , Y: "000101" Record: "110011"
Step 4
Add all the ones and zeros in the record together to obtain the Hamming distance i.e nothing but the distance between the two strings. Distance = 0+0+1+1+0+0+0+0+1+1+1+1 = 4
There is a procedure known as Hamming Code which elaborates the method to operate to detect and correct the errors appear in particular bit of the binary message. It is as follows:
Step 1 : If (2 pow N) of length is the message that is being transmitted from the transmiiter to receiver. Then N number of bits are to be taken as parity bits(check bits, which are used for error detection and correction) and remanining are the data bits which are streamed along with parity bits as a message.
And using the say 7 bit Hamming Code:
P1 which stands for Parity 1 bit, P2 stands for Parity 2 bit and P4 stands for Parity 4 bit.
Message =
This is how the arrangement takes place and b y the Hammer predefined equations,
P1 = D3 D5 D7
P2 = D3 D6 D7
P3 = D5 D6 D7
Values are mapped with Parity and Data bits. And check for the even parity/odd parity
Even Parity = Even Number of 1's/0's within the data bits
Odd Parity = Odd Number of 1's/0's within the data bits
And our objective is to convert those are in Odd parity to even parities in order to detect and correct the errors. Its done by setting corresponding Parity bit as 1. If its already in the even parity then parity bit is set as 0
And finally all parity bits collaborated this way (P4 P2 P1) and to check the corresonfing by knowing binary ewuvalent i.e (P4 P2 P2) base 2. Corresponds to the erroneous bit, And its value is reversed to correct the error.
Q2:
A: Using the above procedure, the answer you would get is (010) base 2.
Q3:
A : Decimal Equivalent of the (010) base 2 is 2
Q4:
A: (0 0 1) base 2
Q5:
A: Decimal Equivalent is 1
Q 6:
A: True
Explaination:
To prove LHS = RHS
n belongs to (0,1,2,........2n)
n pow 2 belongs to (0^2, 1^2, 2^2, 3^2,........... powers of even and odd integers)
iff n pow 2 belongs to (1^2, 3^2, 5^2, ............. powers of odd integers i.e N^2) where N is an odd integer
i.e n^2 belongs to N^2
Square Root on both sides
n belongs to N
i.e N is Odd.
Hence Proved.
Q7:
Ans: As per the statement
if a belongs to even integers
a^2 belongs to even integers
Just like above question this one to be solved deriving examples.
D7 D6 D5 P4 D3 P2 P1