# Boolean Algebra Precedence

the order of precedence for boolean algebra is:

1. Brackets
2. Not
3. And
4. Or

# Boolean Identities

### Using AND

This equation means that the output is determined by the value of A. So if A =0, The output is 0, and vice versa.

Because there is a 0 in this equation, the output of this will always be 0 regardless of the value of A.

The output is determined by A alone in this equation. This can be simplified to just "A".

Here the output will be 0, regardless of A's value. A would have to be 1 and 0 for the output to be 1. This means we can simplify this to just 0.

### Using OR

0 or A can be simplified as just A.

1 or A can be simplified as just 1.

A or A can be simplified as just A.

NOT A or A can be simplified as just 1.

# Boolean Laws

## Commutative Law

The Commutative Law is where equations are the same no matter what way around the letters are written. For example

or

## Associate Law

If all of the symbols are the same it doesn't matter which order the equation is evaluated.

So:

## Distributive Law

The distributive law is these two equations.

This is essentially factorising or expanding the brackets, but you can also remove the common factor:

You can also remove the common factor if you only have 1 term on one side:

if the symbol inside the brackets is a '+' you can add '+0' or if the symbol inside the brackets is '.' you can add '.1'. Doing this will not change the nature of the brackets because 'A' is the same as 'A+0' and is the same as 'A.1'.

## Redundancy Law

Proof :

Proof :

### Law 3:

Proof using distributive law:

So:

So:

### Law 4:

Proof using distributive law:

So:

So:

## Identity Law

This is also in the identities section:

## Negation Law

Just like in any other logic negating a negative is a positive so:

# Solving Boolean Equations

Solving equations is a matter of applying the laws of boolean algrebra, followed by any of the identities you can find:

### Example 1

Take out the common factor C:

,

We know that ,

Therefore, ,

Use identity ,

### Example 2

A.(C+A)

|Use Distributive Law|

->(A.C)+(A.A)

|Use Identity| A.A=A

->(A.C)+A

|This is the same as writing (could straight apply redundancy rule here)|

->(A.C)+(A.1)

|Take out the common factor|

->A.(C+1)

|Use Identity| C+1 = 1

->A

### Example 3

B.(A+ NOT B) REDUNDANCY (A + NOT B) REDUNDANCY

ANSWER = NOT B

### Example 8

D.E+E.D Distributivetive Law D.(E+D) Redundancy Law D

### Example 13

Expand the brackets:

Not B AND B = 0:

Something OR 0 is Something:

### Example 14

(B) + (A.B) Distributive Law. (B + A) . (B + B) Not B cancels out. B + A . 1 = B+A

### Example 19

Distributive:

Identity laws:

#### Alternative

Expanding the brackets

Use of and

Taking X out of the brackets

Use of