1.4.3. Boolean Algebra

Question 1

Logic gates

Answer 1

Take inputs from 1 or more binary switches and reach an output
Can be combined to form a complex circuit

Question 2

Combining logic gates

Answer 2

Make a column in your truth table for each gate, starting from the ones needed first

Question 3

NOT gate

Answer 3

Symbol: One line going into a triangle with a circle at the point and a line coming out
What it does: Turns a 0 to a 1 and a 1 to a 0
Boolean algebra: P = NOT A
Boolean notation: P = ¬ A

Question 4

AND gate

Answer 4

Symbol: Two lines into the long edge of a semi-circle, one coming out
What it does: Only outputs a 1 if both inputs are 1
Boolean algebra: P = A AND B
Boolean notation: A ∧ B

Question 5

OR gate

Answer 5

Symbol: Two lines into the inside of a crescent moon, one out
What it does: If one or both input(s) are 1 it outputs a 1
Boolean algebra: P = A OR B
Boolean notation: P = A V B

Question 6

XOR gate

Answer 6

Symbol: The OR symbol but with another curve the same as the inside of the moon cutting the lines
What it does: Only outputs 1 if one input is 1, not both
Boolean algebra: P = A XOR B
Boolean notation: P = A ⊻ B

Question 7

De Morgan’s First Law

Answer 7

¬(A V B) = ¬A ∧ ¬B

Question 8

De Morgan’s Second Law

Answer 8

¬(A ∧ B) = ¬A V ¬B

Question 9

How to implement de Morgan’s laws

Answer 9

Invert both terms, swap OR and AND and invert the whole result

Question 10

Associative Rule

Answer 10

If the same expression is used on three inputs the brackets can be anywhere or not there
e.g. A V (B V C) = (A V B) V C = A V B V C

Question 11

Commutative Rule

Answer 11

Changing which input comes first makes no difference

E.g. A V B = B V A

Question 12

Distribution

Answer 12

Allows you to multiply out or factorise an expression, done the same as brackets in maths
e.g. X ∧ (Y V Z) = (X ∧ Y) V (X ∧ Z) and vice versa

Question 13

Double negative

Answer 13

¬¬X = X

Question 14

Absorption

Answer 14

When brackets have the first input inside in front of it also with an and and an or inside and out you can eliminate the second variable
X V ( X ∧ Y) = X
X ∧ (X V Y) = X

Question 15

Finding an expression from a karnaugh map

Answer 15

Draw squares/rectangles around groups of 1s, find the expression for each and put around ors

Question 16

Grouping sizes in karnaugh maps

Answer 16

1, 2, 4 or 8

Groups can overlap

Question 17

Order of rows/columns in karnaugh maps

Answer 17

00, 01, 11, 10

Question 18

Wraparound

Answer 18

Combining smaller groups on the edges to make a larger one with a simpler expression

Question 19

Half-adder circuit

Answer 19

Inputs A and B going into a XOR gate to give the digit

The same 2 inputs go into an AND gate for the carry

Question 20

Half-adder vs adder

Answer 20

An adder can add 3 bits so is more practical (A, B and the previous carry bit)

Question 21

Adder circuit

Answer 21

Repeats the half-adder circuit with the output from the XOR and C as the inputs.
The output of the XOR is the digit
Pass an OR with the 2 AND gates for the carry
Don’t need to remember how to draw

Question 22

n Adders

Answer 22

Can add together n pairs of bits with carry bits

Question 23

D-type flip-flop inputs and outputs

Answer 23

Inputs D and the clock

Outputs Q and not Q

Question 24

How does a D-type flip-flop work?

Answer 24

On the rising edge of the clock (when it goes from 0 to 1), Q changes to equal the value of D and not Q the opposite. It will stay like this until the next rising edge (it doesn’t change on a falling edge)

Question 25

D-type flip-flop uses

Answer 25

Storing values (registers or counters)
Static RAM