Boolean Algebra

Any equation must be within the <math> </math> tags. For Boolean alegbra the main issue is how to negate a term like:

$\overline{a}$ or $\overline{\overline{a}+b}$

this can be done by adding the following around any term you wish to negate.:

<math> \overline{} </math>  

$\overline{a}$

is

 <math> \overline{a} </math>

$\overline{\overline{a}+b}$

is

 <math> \overline{\overline{a}+b} </math>.

Identities

AND Identities

$A.1 = A$

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

$0.A = 0$

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

$A.A = A$

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

$A.\overline{A}=0$

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.

OR Identities

$0+A = A$

0 or A can be simplified as just A.

$1+A = 1$

1 or A can be simplified as just A.

$A+A=A$

A or A can be simplified as just A.

$\overline{A}+A=1$

NOT A or A can be simplified as just 1.

Laws

Commutative Law

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

$A+B = B+A$

or

$A.B = B.A$

Associate Law

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

$A+(B+C) = B + (A+C)$

$A+(B+C) = B + (A+C)$

$A+(B+C) = C + (A+B)$

So:

$A.(B.C) = B . (A.C)$

$A.(B.C) = B . (A.C)$

$A.(B.C) = C . (A.B)$

Distributive Law

The distributive law is these two equations.

$A.(B+C) = A.B + A.C$

$A+(B.C) = (A+B).(A+C)$

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

$A.B + A.C = A.(B+C)$

$A+B.A+C = A+(B.C)$

Redundancy Law

Law 1 : $A + A \overline{B} = A + B$

Proof :

$= A + A \overline{B} \\ = (A + \overline{A})(A + B) \\ = 1 . (A + B) \\ = A + B$

Law 2: $A.(\overline{A} + B) = A.B$

Proof :

$= A.(\overline{A} + B) \\ = A.\overline{A} + A.B \\ = 0 + A.B \\ = A.B$

Identity Law

This is also in the identities section:

$A.A = A$

$A+A = A$

Negation Law

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

$\overline{ \overline{A} } = A$

Equations

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

Example 2

$(\overline{A} . B) . (\overline{A} . C)$

Example 6

AB(A + B)(B + B) Original Expression AB(A + B) Complement law, Identity law. (A + B)(A + B) DeMorgan's Law A + BB Distributive law. This step uses the fact that or distributes over and. A Complement, Identity.