Question 1

a) Determine the value of base x if (152)_x= (6A)₁₆ Please show all steps.

b) Convert the followings: (Please show all steps)

i) 0xBED into 3-base representation

ii) 321₇ into 2-base (binary) representation

iii) 1235 into octal representation

iv) 21.21₈ into decimal representation

c) Given a (very) tiny computer that has a word size of 3 bits, what are the lowest value (negative number) and the highest value (positive number) that this computer can represent in each of the following representations?

i) One's complement

ii) Two's complement

iii) Signed Magnitude

Question 2.

a) Are these two combinational circuits equivalent? Provide proof of your claim.

b) Consider the following logic diagram of a combinational circuit where A and B are inputs and X is the output. Two 2-input AND gates, one 2-input OR gate, and two inverters are used in the circuit. It is possible to reduce some of the logic gates without changing the functionality of the circuit. Such component reduction results in higher operating speed (less delay time from input signal transition to output signal transition), less power consumption, less cost, and greater reliability. Construct a logic diagram of a circuit which does have the same function output. Please show the steps.

c) Using basic Boolean algebra identities for Boolean variables x, y, and z, prove that

Please show all steps and mention the identities used.