1. Perform the following using 2's complement: (88)10 + (−37)10. 2. The 2's complement representation of a binary number is 10101100: i) Determine its decimal value. ii) Represent... 1. Perform the following using 2's complement: (88)10 + (−37)10. 2. The 2's complement representation of a binary number is 10101100: i) Determine its decimal value. ii) Represent it in 1's complement form. 3. Using Huntington's postulates prove that a) X + X = X b) X + 1 = 1. 4. Obtain the canonical forms of the Boolean function F(A,B,C) = A'B'C + AB'C + ABC'. 5. Reduce the following function into its minimum SOP Y = A'BCD' + A'BCD + A'BCD' + ABC'D + ABCD + AB'CD. 6. (i) Perform the binary subtraction (1011)2 − (1101)2 using 1's and 2's complements and verify results by direct subtraction. (ii) Perform the subtraction 2210 − 3560 using 9's and 10's complements and verify results by direct subtraction. (iii) Perform the subtraction Hexadecimal numbers B2A and 5C7. (iv) Perform the subtraction Octal numbers 763 and 456. 7. Perform the following operations using 2's complement representation: i) (−34) + (21) ii) (−26) + (−12) iii) (−33) + (−22) iv) (−45) + (−32). Read more

Steps to Solve

1. Performing Binary Subtraction Using 2's Complement

   To calculate the binary subtraction $ (1011_2 - 1101_2) $, we convert the second number to its 2's complement:

   • First, find the 1's complement of $1101_2$:

     $1's \ complement = 0010_2$

   • Then, add $1$ to get the 2's complement:

     $2's \ complement = 0010_2 + 0001_2 = 0011_2$

   • Now add it to $1011_2$:

     $$ \begin{array}{c@{}c@{}c@{}c@{}c@{}c@{}c} & 1 & 1 & 1 & 1 \

     • & 1 & 0 & 1 & 1 \
     • & 0 & 0 & 1 & 1 \ \hline 1 & 0 & 0 & 0 & 0_{(sum)} \ \end{array} $$

   This leads to a final result of $1000_2$, ignoring any overflow since this is a 4-bit operation.

2. Verifying the Result by Direct Subtraction

   Confirm the subtraction directly:

   • Converting to decimal: $1011_2 = 11_{10}$, and $1101_2 = 13_{10}$
   • So, $11 - 13 = -2$, and in binary, $-2$ is represented by $1110_2$ (for 4 bits).

3. Representing Decimal Values in 2's Complement

   For the binary number $10101100$, to find the 2's complement:

   • First, calculate the 1's complement:

     $1's \ complement = 01010011$

   • Then add $1$:

     $2's \ complement = 01010011 + 00000001 = 01010100$

4. Convert to Decimal Value

   The decimal equivalent of $10101100_2$:

   $$ = 1 \times 2^7 + 0 \times 2^6 + 1 \times 2^5 + 0 \times 2^4 + 1 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 0 \times 2^0 $$ $$ = 128 + 32 + 8 + 4 = 172_{10} $$

5. Finding 1's Complement

   The 1's complement of $10101100$ is simply flipping all bits:

   $$ 01010011 $$