# Applied Electronics Argumentation Boolean Identities

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15 October 19:18

There are several laws that can be acclimated to abridge or adapt Boolean expressions. This page will explain them, will account them for simple reference.

Axioms are propositions that are accounted accessible and accordingly are not appropriate to be accepted - in actuality they cannot be accepted because they are authentic to be true by the anatomy of the algebra. They anatomy the foundation of the blow of boolean algebra by defining the two values, 1 and 0 and the three operators, AND, OR and NOT.

X

e1, ,X=0||$mboxX$

e0, ,X=1

|-

|2||$mboxX=0,\; ,\; overline\; =1$||$mboxX\; =\; 1,\; ,\; overline=0$

|-

|3||$0\; cdot\; 0\; =\; 0$||$1\; +\; 1\; =\; 1\; ,$

|-

|4||$1\; cdot\; 1\; =\; 1$||$0\; +\; 0\; =\; 0\; ,$

|-

|5||$1\; cdot\; 0\; =\; 1\; cdot\; 0\; =\; 0$||$1\; +\; 0\; =\; 0+1=\; 1\; ,$

|}

DeMorgans Law is a actual able apparatus for alignment or ungrouping analytic statements. It basically states that either analytic action AND or OR may be replaced by the other, accustomed assertive changes to the equation. It is usually bidding as two audible identities. First is the following:

::$overline=overline\; A\; cdot\; overline\; B$

At first, it may not be bright how this aftereffect is found, but if you attending at the first Venn Diagram to the left, it is clear. Brainstorm a capricious alleged X, which is authentic ::$X=overline$

This agency that X is NOT in accumulation A OR B, so it haveto be in the dejected region. Now, this is the aforementioned as adage that X is NOT in A AND aswell NOT in B. This agency that we can aswell ascertain X as:

::$X=overline\; A\; cdot\; overline\; B$

This can aswell be apparent by cartoon up a accuracy table: alive in from anniversary end of the table, we can see that the two accord identical results:

The additional announcement is basically the aforementioned but with exchanged symbols

::$overline=overline\; A\; +\; overline\; B$

If we do the aforementioned ambush with X again, we see that it haveto NOT be in both A AND B. Accordingly it haveto be in the ablaze blue, aphotic dejected or red regions, and not in the centre. We can now say that X is either not in A, not in B or not in either. This is the the aforementioned as adage that X is NOT in A OR NOT in B (using the analytic affectionate of or). Therefore,

::$X=overline\; A\; +\; overline\; B$

Again, this can be apparent by a accuracy table:

An simple way to bethink the laws accustomed aloft is: Breach the line, change the sign. With these two rules, it can be apparent that any analytic account can be afflicted using the afterward method:

#Compliment all agreement in the expression

#Change all ANDs to ORs and all ORs to ANDs

#Invert the result

There are several laws that can be acclimated to abridge or adapt Boolean expressions. This page will explain them, will account them for simple reference.

Axioms are propositions that are accounted accessible and accordingly are not appropriate to be accepted - in actuality they cannot be accepted because they are authentic to be true by the anatomy of the algebra. They anatomy the foundation of the blow of boolean algebra by defining the two values, 1 and 0 and the three operators, AND, OR and NOT.

X

e1, ,X=0||$mboxX$

e0, ,X=1

|-

|2||$mboxX=0,\; ,\; overline\; =1$||$mboxX\; =\; 1,\; ,\; overline=0$

|-

|3||$0\; cdot\; 0\; =\; 0$||$1\; +\; 1\; =\; 1\; ,$

|-

|4||$1\; cdot\; 1\; =\; 1$||$0\; +\; 0\; =\; 0\; ,$

|-

|5||$1\; cdot\; 0\; =\; 1\; cdot\; 0\; =\; 0$||$1\; +\; 0\; =\; 0+1=\; 1\; ,$

|}

DeMorgans Law is a actual able apparatus for alignment or ungrouping analytic statements. It basically states that either analytic action AND or OR may be replaced by the other, accustomed assertive changes to the equation. It is usually bidding as two audible identities. First is the following:

::$overline=overline\; A\; cdot\; overline\; B$

At first, it may not be bright how this aftereffect is found, but if you attending at the first Venn Diagram to the left, it is clear. Brainstorm a capricious alleged X, which is authentic ::$X=overline$

This agency that X is NOT in accumulation A OR B, so it haveto be in the dejected region. Now, this is the aforementioned as adage that X is NOT in A AND aswell NOT in B. This agency that we can aswell ascertain X as:

::$X=overline\; A\; cdot\; overline\; B$

This can aswell be apparent by cartoon up a accuracy table: alive in from anniversary end of the table, we can see that the two accord identical results:

The additional announcement is basically the aforementioned but with exchanged symbols

::$overline=overline\; A\; +\; overline\; B$

If we do the aforementioned ambush with X again, we see that it haveto NOT be in both A AND B. Accordingly it haveto be in the ablaze blue, aphotic dejected or red regions, and not in the centre. We can now say that X is either not in A, not in B or not in either. This is the the aforementioned as adage that X is NOT in A OR NOT in B (using the analytic affectionate of or). Therefore,

::$X=overline\; A\; +\; overline\; B$

Again, this can be apparent by a accuracy table:

An simple way to bethink the laws accustomed aloft is: Breach the line, change the sign. With these two rules, it can be apparent that any analytic account can be afflicted using the afterward method:

#Compliment all agreement in the expression

#Change all ANDs to ORs and all ORs to ANDs

#Invert the result

Tags: electronics overline, cdot, logical, boolean, table, identities, , logic boolean identities, electronics logic boolean, practical electronics logic, |

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