1. Add (-9<sub>10</sub>) + (-11<sub>10</sub>) in binary using 2's complement representation. Use a word length of 6 bits (including the sign bit) and indicate whether an overflow occurs. Convert the result of this addition back to decimal. 2. Minimize (X+Y)(X'+Z)Z'Y' + X(Y+Z) + (ZYX+Y)(ZYX+XZ) 3. Without simplifying, use DeMorgan's law to find the inverse of (X+Y)(X'+Z)Z'Y' + X(Y+Z) + (ZYX+Y)(ZYX+XZ) 4. Convert these expressions to both sum of products and product of sums, minimizing where possible abc+(d+e)c'+abc' a(b+c(d+e))+d
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To convert a negative number to binary using 2's complement representation, we first convert the absolute value of the number to binary, then invert all the bits, and finally add 1 to the result. -91° in binary: 91° in binary = 01011011 Invert all the bits: Show more…
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2. Convert the following unsigned binary numbers to decimal and hexadecimal. Show your work. a. 11110 b. 11010 c. 100100011 3. Convert the following two's complement binary numbers to decimal and hexadecimal. Show your work. a. 1001011 b. 011110 4. Derive the truth table for the following Boolean function. F (x, y, z) = x'y'z' + x'yz + xy'z' + xyz 5. Derive the Boolean function F for the following truth table. Show your work. | a | b | c | F | |---|---|---|---| | 0 | 0 | 0 | 0 | | 0 | 0 | 1 | 0 | | 0 | 1 | 0 | 1 | | 0 | 1 | 1 | 1 | | 1 | 0 | 0 | 0 | | 1 | 0 | 1 | 0 | | 1 | 1 | 0 | 1 | | 1 | 1 | 1 | 0 | 6. Use Boolean algebra to show that z + y' + yz' = 1. Show your work. 7. Write the following equations as a Boolean functions in a canonical form: a. F(x, y, z) = Σ (1, 3, 7) b. F(x, y, z) = Π (1, 3, 7) 8. Given F = xy'z' + xy'z + xyz' + xyz, write the equation for F' using the product of sums format, and the sum of products format.
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Convert the following decimal numbers to 6-bit two's complement binary numbers and add them. Indicate whether or not the sum overflows a 6-bit result. (a) 16 + 9 (b) 27 + 31 (c) -4 + 19 (d) 3 + (-32) (e) -16 + (-9) (f) -27 + (-31)
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Multiply each of the following pairs of signed 2's-complement numbers using the Booth algorithm. In each case, assume that A is the multiplicand and B is the multiplier. a) A = 010111 and B = 110110 b) A = 110011 and B = 101100 c) A = 001111 and B = 001111 Repeat Problem 3 using bit-pair recording of the multiplier.
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