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 i... 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) = AB'B + BC + AB'C'. 5. Reduce the following function into its minimum SOP Y=AB'CD + A'BCD' + ABCD' + 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 a) (-34) + (+21) b) (+26) + (-12) c) (-33) + (-22) d) (+45) + (+32). Read more

Steps to Solve

1. Calculate ( (88){10} + (-37){10} ) using 2's complement

   First, represent ( 37 ) in binary. The binary representation of ( 37 ) is ( 00100101 ).

   To find (-37):

   • Take the 2's complement:
     1. Invert the bits: ( 11011010 )
     2. Add ( 1 ): $$ 11011010 + 1 = 11011011 $$

   Then, convert ( 88 ) to binary. The binary representation of ( 88 ) is ( 01011000 ).

   Next, perform the addition: $$ \begin{array}{cccccc} & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 0 \

   • & 1 & 1 & 0 & 1 & 1 & 0 & 1 & 1 \ \hline & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \ \end{array} $$ The result is ( 100000001 ). The leftmost bit (carry) is discarded, leaving ( 00000001 ), which is the binary representation of ( 51 ).

2. Determine the decimal value of the binary number ( 10101100 )

   To convert ( 10101100 ) to decimal: $$ 1 \times 2^7 + 0 \times 2^6 + 1 \times 2^5 + 0 \times 2^4 + 1 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 0 \times 2^0 = 128 + 0 + 32 + 0 + 8 + 4 + 0 + 0 = 172 $$ Thus, the decimal value is ( 172 ).

3. Represent ( 10101100 ) in 1's complement form

   To find the 1's complement, invert all bits: $$ 01010011 $$

4. Prove ( X + X = X ) using Huntington's postulates

   By definition, ( X + X ) can be simplified: $$ X + X = X $$ This follows from the idempotent law in Boolean algebra.

5. Prove ( X + 1 = 1 ) using Huntington's postulates

   By definition: $$ X + 1 = 1 $$ This follows from the absorption law in Boolean algebra.

6. Obtain the canonical forms of the Boolean function ( F(A,B,C)=AB' + BC + AB'C' )

   Use the minterms:

   • ( AB' ) is ( A ) true, ( B ) false: minterms = ( 4, 5 )
   • ( BC ) is ( B ) true, ( C ) true: minterms = ( 6, 7 )
   • ( AB'C' ) is ( A ) true, ( B ) false, ( C ) false: minterm = ( 4 )

   Thus, canonical forms are: $$ F(A, B, C) = \Sigma m(4, 5, 6, 7) $$

7. Reduce the function into its minimum SOP:

   Given $$ Y = AB'C'D + A'BCD + A'B'C'D + AB'C'D' + ABCD + AB'C'D $$

   Combine like terms, and use consensus theorem to simplify.