Boolean Algebra

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

[math] \overline{a}[/math] or [math] \overline{\overline{a}+b}[/math]

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

<math> \overline{} </math>  

[math] \overline{a}[/math]

is

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

[math] \overline{\overline{a}+b}[/math]

is

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

Identities

AND Identities

[math] A = A [/math]

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

[math] 0.A = 0 [/math]

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

[math] A.A = A[/math]

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

[math] A.\overline{A}=0 [/math]

The output of this equation will always be 0 as A will need to be both 1 and 0 at the same time for this logic to return a 1.

[math] A.B = B.A [/math]

This equation can be written like this as it does not matter which way around the values are entered. The output will be the same.

[math] A.(B.C) = (A.B).C [/math]

This equation can be written like this as all values must be equal to 1 for the output to be 1. Therefore the order in which it is written does not matter.

[math] A+(B.C) = (A.B).(A+C) [/math]

OR Identities

[math] 0+A = A [/math]

"If 0 or A goes in, A is the output"

[math] 1+A = 1 [/math]

"If 1 or A goes in, 1 is the output"

[math] \overline{A}+1=1[/math]

"If NOT A or 1 goes in, the output is 1"

[math] \overline{A}+A=1[/math]

"If NOT A or A goes in, the output is 1"

Laws

Commutative Law

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

[math] A+B=B+A [/math]

or

[math] A.B=B.A [/math]

Laws

Distributive Law

The distributive law is these two equations.

[math] A.(B+C) = A.B + A.C [/math]

[math] A+(B.C) = (A+B).(A+C) [/math]

Redundancy Law

[math] \overline{A} = \overline{A} [/math] or [math] \overline{\overline{A}} = A [/math]

Identity Law

[math] A+A = A [/math]

Equations

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