Section 8 - Boolean Algebra

Question 1

Define what an ‘NOT gate’ / Negation represents

Answer 1

Inverts the input - 0 to 1 and 1 to 0

Question 2

Define what an ‘AND gate’/Conjunction represents

Answer 2

The output of A AND B is 1 only if the input A and input B are both 1

Question 3

Define what an ‘OR gate’/Disjunction represents

Answer 3

The output of A OR B is 1 if either the input for A or B is 1

Question 4

Define what an ‘XOR gate’ /Exclusive disjunction represents

Answer 4

The output of A XOR B is 1 if only one of A or B is 1

Question 5

Define ‘de Morgan’s Laws’

Answer 5

Any Boolean function can be converted to one which uses only NAND functions or only NOR functions.

Question 6

What is de Morgan’s first Law?

Answer 6

¬(A v B) = ¬A ∧ ¬B

X = ¬(A v B)

X = ¬A ∧ ¬B

Question 7

What is de Morgan’s second Law?

Answer 7

¬(A ∧ B) = ¬A v ¬B

X = ¬(A ∧ B) = ¬A v ¬B

X = ¬A v ¬B

Question 8

What are the general rules of Boolean Algebra

Answer 8

X ∧ 0 = 0

X ∧ X = X

X ∧ 1 = X

X ∧ ¬X = 0

X v 0 = X

X v 1 = 1

X v X = X

X v ¬X = 1

Question 9

What are the two commutative rules of Boolean Algebra?

Answer 9

X ∧ Y = Y ∧ X

X v Y = Y v X

Question 10

What are the two associative rules of Boolean Algebra?

Answer 10

X ∧ (Y ∧ Z) = (X ∧ Y) ∧ Z

Question 11

What are the two associative rules of Boolean Algebra?

Answer 11

X ∧ (Y ∧ Z) = (X ∧ Y) ∧ Z

X v (Y v Z) = (X v Y) v Z

Question 12

What are the two distributive rules of Boolean Algebra?

Answer 12

X ∧ (Y v Z) = (X ∧ Y) v (X ∧ Z)

(X v Y) ∧ (W v Z) = (X ∧ W) v (X ∧ Z) v (Y ∧ W) v (Y ∧ Z)

Question 13

What are the two rule of absorption in Boolean Algebra?

Answer 13

X v (X ∧ Y) = X

X ∧ (X v Y) = X

Question 14

What is the rule for double negation in Boolean Algebra?

Answer 14

X = ¬ ¬X

Question 15

Define what is meant by the term ‘half-adders’

Answer 15

Takes two input of two bits and gives a two-bit output as the correct result of an addition of the two inputs.

Question 16

Disadvantage of half-adder

Answer 16

Only has two inputs meaning unlike a full adder it cannot use the carry from a previous addition as a third input in order to add n-bit numbers.

Question 17

Define what is meant by the term ‘full-adder’

Answer 17

Combines two half-adders to add three bits together including inputs A and B, and a carry bit C. Additional Or gate to output the carry bit.

Question 18

Define what is meant by the term ‘flip flop’

Answer 18

Elemental sequential logic circuit that can store one bit and flip between two states, 0 and 1. Two inputs a control signal labelled D and a clock signal.

Question 19

What is the role of the clock in a flip flop?

Answer 19

Changes states at regular time intervals, needed to synchronise the change of state of flip flop circuits.

Question 20

What is a D-type flip-flop?

Answer 20

Positive edge-triggered flip-flop can only change output value when the clock is at a rising or positive edge.

Question 21

Why are flip-flop circuits important?

Answer 21

Used as a memory cell to store the state of a bit.

Question 22

How does the output Q change in accordance with the input D in a flip-flop?

Answer 22

Q only changes to match the input D in response to the rising edge on the clock signal. Q stores the value D by up to one clock cycle if the state does not change.