A Level Computer Science Paper 1: 1.4.3. Boolean Algebra

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

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

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

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

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

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

¬(A V B) = ¬A ∧ ¬B

¬(A ∧ B) = ¬A V ¬B

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

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

Changing which input comes first makes no difference

E.g. A V B = B V A

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

¬¬X = X

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

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

1, 2, 4 or 8

Groups can overlap

00, 01, 11, 10

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

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

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

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

Can add together n pairs of bits with carry bits

Inputs D and the clock

Outputs Q and not Q

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)

Storing values (registers or counters)

Static RAM