[SOLVED] BLG 231E - Digital Circuits Assignment 1

Part 1 – Computer Arithmetic

1. A and B are two 8-bit binary integers, and B = 1101 1001. For the operation A – B, answer the following questions:
   1. If A and B are signed binary integers, what are the i) largest and ii) smallest decimal values of A that yield valid results (that can be represented using 8 bits) after the operation? Explain your answer briefly.
   2. Write the binary representation for the largest value of the signed A you found in (a.i). Carry out the binary operation A – B using 2’s complement, and show that the result is valid using binary numbers only.
   3. If A and B are unsigned binary integers, what are the i) largest and ii) smallest binary values of A that yield valid results after the operation? Explain your answer briefly.

2. A and B are two 8-bit, signed, binary integers, and A = 1011 1100. If we perform the operation A + B ,
   1. What are the i) largest and ii) smallest decimal values of B that yield valid results after the operation? Explain your answer briefly.
   2. Write the binary representation for the smallest value of the signed B you found in (a.ii). Perform the binary operation A + B, and show that the result is valid using binary numbers only.

Part 2 – Boolean Algebra

3. Simplify the following logic expressions using axioms, properties, and theorems of Boolean algebra.

1. 𝐸(𝑎, 𝑏, 𝑐) = 𝑎𝑏̅𝑐 + 𝑎𝑏𝑐̅ + 𝑎𝑏𝑐 + 𝑎̅𝑏𝑐
2. 𝐸(𝑎, 𝑏, 𝑐, 𝑑) = 𝑎̅𝑏𝑑̅ + 𝑏𝑐𝑑 + 𝑎𝑏𝑐̅ + 𝑎𝑏̅𝑑 + 𝑏𝑐̅𝑑̅ + 𝑎𝑑 + 𝑎̅𝑏𝑐