Difference between revisions of "Boolean Algebra"
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==Example 2== | ==Example 2== | ||
<math> (\overline{A} + A) . (\overline{A} + C) </math> | <math> (\overline{A} + A) . (\overline{A} + C) </math> | ||
− | + | OR Identity | |
<math>(\overline{A} + A) = 1</math> | <math>(\overline{A} + A) = 1</math> | ||
− | + | AND Identity | |
<math>= 1.(\overline{A} + C)</math> | <math>= 1.(\overline{A} + C)</math> | ||
Revision as of 10:56, 9 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
0 or A can be simplified as just A.
1 or A can be simplified as just A.
A or A can be simplified as just A.
NOT A or A can be simplified as just 1.
Laws
Commutative Law
The Commutative Law is where equations are the same no matter what way around the letters are written. For example
or
Associate Law
If all of the symbols are the same it doesn't matter which order the equation is evaluated.
So:
Distributive Law
The distributive law is these two equations.
This is essentially factorising or expanding the brackets, but you can also:
Redundancy Law
Law 1 :
Proof :
Law 2:
Proof :
Identity Law
This is also in the identities section:
Negation Law
Just like in any other logic negating a negative is a positive so:
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
OR Identity AND Identity
Example 3
Distributive:
Identity laws:
Example 4
Expression Rule(s) Used C + BC Original Expression C + (B + C) DeMorgan's Law. (C + C) + B Commutative, Associative Laws. T + B Complement Law. T Identity Law.
Example 5
Simplify: AB(A + B)(B + B): Expression Rule(s) Used AB(A + B)(B + B) Original Expression AB(A + B) Complement law, Identity law. (A + B)(A + B) DeMorgan's Law A + BB Distributive law. This step uses the fact that or distributes over and. It can look a bit strange since addition does not
distribute over multiplication.
A Complement, Identity.
Example 6
Original Expression
Complement law, Identity law.
DeMorgan's Law
Distributive law. This step uses the fact that or distributes over and.
Complement, Identity.
Example 7
SIMPLIFY
Complement, Identity.
Commutative, Distributive.
Associative, Idempotent.
Distributive.
Idempotent, Identity, Distributive.
Identity, twice.
Example 8
Simplify
solution
Example 9