Difference between revisions of "Boolean Algebra"
(→AND Identities) |
Jmiester26 (talk | contribs) (→Redundancy Law) |
||
Line 74: | Line 74: | ||
==Redundancy Law== | ==Redundancy Law== | ||
− | <math> \overline{A} = \overline{A} </math> | + | Law 1 : |
− | + | <math> A + A \overline{B} = A + B </math> | |
− | <math> \overline{\overline{A}} = A </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== | ==Identity Law== |
Revision as of 08:40, 8 May 2018
Any equation must be within the <math> </math> tags. For Boolean alegbra the main issue is how to negate a term like:
or
this can be done by adding the following around any term you wish to negate.:
<math> \overline{} </math>
is
<math> \overline{a} </math>
is
<math> \overline{\overline{a}+b} </math>.
Contents
Identities
AND Identities
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.
OR Identities
If 0 or A goes in, A is the output
If 1 or A goes in, 1 is the output
If NOT A or 1 goes in, the output is 1
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
orLaws
Associate Law
Distributive Law
The distributive law is these two equations.
Redundancy Law
Law 1 :
Proof :
Law 2:
Proof :
Identity Law
Negation Law
Equations
Solving equations is a matter of applying the laws of boolean algrebra, followed by any of the identities you can find: