Boolean Algebra
Contents
- 1 Boolean Algebra Precedence
- 2 Boolean Identities
- 3 Boolean Laws
- 4 Solving Boolean Equations
Boolean Algebra Precedence
the order of precedence for boolean algebra is:
- Brackets
- Not
- And
- Or
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 remove the common factor:
You can also remove the common factor if you only have 1 term on one side:
Redundancy Law
Law 1 :
Proof :
Law 2:
Proof :
Law 3:
Proof using distributive law:
So:
So:
Law 4:
Proof using distributive law:
So:
So:
Identity Law
This is also in the identities section:
Negation Law
Just like in any other logic negating a negative is a positive so:
Solving 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
Take out the common factor C:
,
We know that
,Therefore,
,Use identity
,Answer =
Example 2
A.(C+A)
|Use Distributive Law|
->(A.C)+(A.A)
|Use Identity| A.A=A
->(A.C)+A
|This is the same as writing (could straight apply redundancy rule here)|
->(A.C)+(A.1)
|Take out the common factor|
->A.(C+1)
|Use Identity| C+1 = 1
->A
Example 3
B.(A+ NOT B) REDUNDANCY (A + NOT B) REDUNDANCY
ANSWER = NOT B
Example 4
Example 5
Example 6
Example 7
B.C
Example 8
D.E+E.D Distributivetive Law D.(E+D) Redundancy Law D
Example 9
Example 10
Example 11
Example 12
Example 13
Example 14
(B) + (A.B) Distributive Law. (B + A) . (B + B) Not B cancels out. B + A . 1 = B+A
Example 15
Example 16
= NOT A
Because the centre symbol is an OR gate and neither of the outputs of the and gates is A therefore by redundancy rule it is NOT A.
Example 17
Example 18
Example 19
Distributive:
Identity laws:
Alternative
Expanding the brackets
Use of and
Taking X out of the brackets
Use of