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Boolean Algebra Tutorial 1 - Printable Version

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Boolean Algebra Tutorial 1 - Hidden Dragon - 01-29-2011 09:53 PM

Welcome to my boolean algebra tutorial series. This is going to be part one of a multi-part series. I'm not sure yet how many there will be.

What is boolean algebra? Well, boolean algebra is a deductive mathematical system closed over the values 0 and 1 (false and true respectively). Boolean logic is the basis for computation in binary computer systems. Any algorithm or electronic circuit can be represented by a system of boolean equations.

Boolean algebra uses binary operators. A binary operator accepts a pair of boolean inputs (1 or 0, true or false) and produces a single output based on them (again 1, 0, true, false). For example, the boolean AND operator accepts two inputs, ANDs them, and produces the result.

Postulates of boolean algebra:
Closure: the boolean system is closed with respect to a binary operator if for every pair of boolean values it produces a single boolean result.
example: Logical AND is closed because it accepts only boolean operands and produces only boolean results.
Commutativity: A binary operator * is commutative if A*B = B*A for all possible values of A and B.
Associativity: A binary operator * is associative if (A*B)*C = A*(B*C) for all possible values of A, B, and C.
Distribution: Two binary operators * and ` are distributive if A*(B`C) = (A*B)`(A*C) for all A, B, and C values.
Identity: A boolean value I is an identity element with respect to another operator * if A*I = A
Inverse: A boolean value I is the inverse element with repect to an operator * if A*I=B and B does not = A (meaning B is the opposite of A).


Now that we have the postulates covered, lets cover the usual operators.
The symbol * represents the logical AND operation. A*B is the result of logically ANDing boolean values A and B.
The symbol + represents the logical OR operation. A+B is the result of logically ORing boolean values A and B.
The symbol ' is logical NOT. ` is unary meaning that it has only one operand. A' is the result of NOTing A boolean value A.


The order of operations in boolean algebra are also important. The order goes parenteses (), NOT ', AND *, then OR +.


I really lied about the postulates. Those weren't all of them. These postulates cover specific operators though so they are a bit different.
Boolean Algebra is closed under AND, Or, and NOT
The identity element with repect to * is 1 and to + is 0.
There is not identity element with '
The * and + operators are commutative
* and + are distributive with repect to each other
Ex. A*(B+C)=(A*B)+(A*C) and A+(B*C)=(A+B)*(A+C)
For every value of A there is a A' so A*A'=0 and A+A'=1.

Those are all the postulates. We do, however, also have theorums that we have to cover. There are 16 important theorums.
1. A+A=A
2. A*A=A
3. A+0=A
4. A*1=A
5. A*0=0
6. A+1=1
7. (A+B)'=A'*A'
8. (A*B)'=A'+B'
9. A+A*B=A
10. A*(A+B)=A
11. A+A'*B=A+B
12. A'*(A+B')=A'*B'
13. A*B+A*B'=A
14. (A'+B')*(A'+B)=A'
15. A+A'=1
16. A*A'=0

Those are all 16 theorums that we're going to need. I know it can be a lot to take in at once so read over and review this so far. Next up we're going to be building boolean truth tables.


RE: Boolean Algebra Tutorial 1 - Ironside - 01-29-2011 10:17 PM

The exact same thing we have this year in 2nd grade of middle school, except that we never took those 16 postulates. We only have 7 logical functions:

- Not,
- And,
- Or,
- Nand,
- Nor,
- Xor,
- Xnor.


RE: Boolean Algebra Tutorial 1 - Hidden Dragon - 01-30-2011 02:23 AM

Well NAND, NOR, and XNOR are actually different gates in electrical engineering so I'll be talking about them in later tutorials.

I don't include XOR in there because you can just OR an equation and then OR the inverses, then finally AND both of those answers.


RE: Boolean Algebra Tutorial 1 - Ironside - 01-30-2011 11:06 AM

Alright then.


RE: Boolean Algebra Tutorial 1 - Jcapers - 10-20-2012 03:09 AM

AB+A(B+C)+B(B+C)=


RE: Boolean Algebra Tutorial 1 - Jcapers - 10-20-2012 03:11 AM

(D+E+F)(D+E+F)F NOT