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]

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

[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

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

Proof :

[math]= A + A \overline{B} \\ = (A + \overline{A})(A + B) \\ = 1 . (A + B) \\ = A + B [/math]

Law 2: [math] A.(\overline{A} + B) = A.B[/math]

Proof :

[math]= A.(\overline{A} + B) \\ = A.\overline{A} + A.B \\ = 0 + A.B \\ = A.B [/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: