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Given a (very) tiny computer that has a word size of 6 bits, what are the smallest negative numbers and the largest positive numbers that this computer can represent in each of the following representations.

i) One's complement

ii) Two's complement

Consider the following logic diagram of a combinational circuit where A, B, and C are inputs and Q is the output. Three 2-input AND gates and two 2-input OR gates 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 with only two logic gates (instead of five). Please show the steps.

b) Using basic Boolean algebra identities for Boolean variables A, B and C, prove that ABC+ ABC' + AB'C + A'BC = AB + AC + BC. Please show all steps and mention the identities used.

- a) Determine the value of basexif (211)
_{x}_{ }= (6A)_{16}[5 marks]

The base radix x in the above equation can be determined in the following way

- Initially the numbers at the both side are converted into the common number system
- The numbers of the same system are then equated to determine the radix x

**Step1:** The numbers are converted into decimal form

(211) _{x}_{ }= (2 * x^{2} + 1*x^{1} +1*x^{0})

= (2x^{2} + x+1)_{10}

(6A) _{16} = (6 * 16^{1}) + (10 * 16^{0})

= (106)_{10}

**Step2:** Decimal forms are equated

2x^{2} + 1x+1 =106

2x^{2} + x-105=0

2x^{2} -14x+15x -105=0

2x(x-7) +15 (x-7) = 0

(2x+15)(x -7)=0

x=7 or x=7.5

Thus considering only the integer values we get x=7

- b) Convert the followings: [3+3=6 marks]
- i) 0xBAD into a decimal number
- i) 0xBAD is a hexadecimal number

A number in one decimal form can be converted to decimal number by multiplying each digit in the number system with their radix power of their location.

Hence

(0xBAD) _{16 }= (B*16^{2 }+ A*16^{1 }+ D*16^{0})

= (11* 256 + 10 * 16 + 13 * 1)

= 2989_{10}

588_{10} into a 3-base number

Decimal numbers are converted to 3-base number by repeatedly dividing the number/quotient with the number 3

3|588

3 |196 – 0

3 |65 – 1

3 |21 – 2

3 |7 - 0

2 -1

Thus (588)_{10} = (210210)_{3}

- c) Given a (very) tiny computer that has a word size of 6 bits, what are the smallest negative numbers and the largest positive numbers that this computer can represent in each of the following representations? [3 +3 = 6 marks]
- i) One's complement

For a n-bit computer system, one’s complement number system represents numbers in the range from − (2^{n−1}−1) to 2^{n−1}−1

Thus for a computer with word size 6 which represents number in one’s complement form,

The smallest negative number that can be represented is – (2^{6-1}-1) = - (32-1) = -31

The largest positive number that can be represented is (2^{6-1}-1) = (32-1) = 31

- ii) Two's complement

For a n-bit computer system, two’s complement number system represents numbers in the range from − 2^{n−1} to 2^{n−1}−1

Thus for a computer with word size 6 which represents number in two’s complement form,

The smallest negative number that can be represented is – (2^{6-1}) = - 32

The largest positive number that can be represented is (2^{6-1}-1) = (32-1) = 31

** **a) Consider the following logic diagram of a combinational circuit where A, B, and C are inputs and Q is the output. Three 2-input AND gates and two 2-input OR gates 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 with only two logic gates (instead of five). Please show the steps.

The number of gates used in the combinational circuit can be reduced by minimizing the equation represented by the circuit. Thus the functionality of the circuit does not change too.

In the above combinational circuit A, B and C are the input lines and Q acts as the output line.

The Boolean equation represented by the above combinational circuit.

Q = AB + ((B+C) BC)

The above equation is minimized by applying Boolean laws

AB + ((B+C) BC) = (AB) + (BBC+CBC) [By Distributive law]

= (AB) + (BC+BC) [By Idempotent law on product]

= AB +BC [By Idempotent law on sum]

= (A+C) B [By Distributive law]

Thus the above minimized equation contains only two symbol which in turn requires only two gates.

The equivalent minimized combinational circuit diagram with two logic gates is given below

- b) Using basic Boolean algebra identities for Boolean variablesA, B and C, prove that ABC+ ABC' + AB'C + A'BC = AB + AC + BC. Please show all steps and mention the identities used. [5 marks]

**To Prove:** ABC+ ABC' + AB'C + A'BC = AB + AC + BC

**Proof:**

Taking right hand side expression

AB + AC + BC = AB.1+AC.1+BC.1 [By Applying Identity law]

= AB(C+C') +AC (B+B') + BC (A+A') [By Applying Inverse law]

= ABC+ABC'+ ABC+AB'C +ABC+A'BC [By Applying Distributive Law]

= ABC+ABC +ABC'+ AB'C +ABC +A'BC [Rearranging the terms in the Boolean expression]

= ABC+ABC'+ AB'C +ABC+ A'BC [By Applying Idempotent law]

= ABC+ABC +ABC'+ AB'C +A'BC [Rearranging the terms in the Boolean expression]

= ABC +ABC'+ AB'C +A'BC

Hence Proved

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