Difference between revisions of "Boolean Algebra"
Jmiester26 (talk | contribs) (→Example 11) |
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<nowiki> <math> \overline{\overline{a}+b} </math></nowiki>. | <nowiki> <math> \overline{\overline{a}+b} </math></nowiki>. | ||
| − | =Identities= | + | =Boolean Identities= |
| − | ==AND | + | ===Using AND== |
<math> A.1 = A </math> | <math> A.1 = A </math> | ||
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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. | 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 | + | ===Using OR=== |
<math> 0+A = A </math> | <math> 0+A = A </math> | ||
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NOT A or A can be simplified as just 1. | NOT A or A can be simplified as just 1. | ||
| − | =Laws= | + | =Boolean Laws= |
==Commutative Law== | ==Commutative Law== | ||
The Commutative Law is where equations are the same no matter what way around the letters are written. For example | The Commutative Law is where equations are the same no matter what way around the letters are written. For example | ||
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<math> \overline{ \overline{A} } = A </math> | <math> \overline{ \overline{A} } = A </math> | ||
| − | =Equations= | + | =Boolean Equations= |
Solving equations is a matter of applying the laws of boolean algrebra, followed by any of the identities you can find: | Solving equations is a matter of applying the laws of boolean algrebra, followed by any of the identities you can find: | ||
Revision as of 14:44, 29 June 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>.
Boolean Identities
=Using AND
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.
Using OR
0 or A can be simplified as just A.
1 or A can be simplified as just 1.
A or A can be simplified as just A.
NOT A or A can be simplified as just 1.
Boolean 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:
Boolean Equations
Solving equations is a matter of applying the laws of boolean algrebra, followed by any of the identities you can find:
Example 1
Use Negation Law
Use De Morgan's Law
Use Associate Law
Example 2
OR Identity
AND Identity
Simplify
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
Expanding the brackets
Use of and
Taking X out of the brackets
Use of
Example 9
after distributive law applied
Example 10
Simplify:
Complement law, Identity law
DeMorgan's law
Distributive law
Complement, Identity
Example 11
Example 12
Example 13
Simplify
First, using De Morgan’s Law simplify it to
This can then be simplified to
Then using De Morgan’s Law again, you get
Then there are double negatives, so you end up with
Example 14
Simplify:
using the Distributive Law
using the identity
using the identity
Fully simplified the equation
Example 15
Simplify the following:
Using De Morgan's law we can put the equation into another form:
1. Change All operations,
2. Negate letters:
3. Negate the whole expression:
Using the identity law, we can then simplify this expression to:
Example 16
For this example the Boolean Algebra itself is shown above while the description/the rules that have been used are listed below it:
Original expression
Idempotent (AA to A), then Distributive, used twice.
Complement, then Identity. (Strictly speaking, we also used the Commutative Law for each of these applications.)
Distributive, two places.
Idempotent (for the A's), then Complement and Identity to remove BB.
Commutative, Identity; setting up for the next step.
Distributive.
Identity, twice (depending how you count it).
Commutative.
Distributive.
Complement, Identity.
