Boolean Algebra

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Revision as of 08:33, 8 May 2018 by Ianjohnson (talk | contribs) (Commutative Law)
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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

Laws

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

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]

Associate Law

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]

Negation Law

Equations

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

Example 1

Example 2

Example 3

Example 4

Example 5

Example 6

Example 7