Keywords: Boolean algebra, XorGate, logic gates
Journal of Computer Sciences and Applications, 2013 1 (1),
pp 1416.
DOI: 10.12691/jcsa113
Received December 26, 2012; Revised February 04, 2013; Accepted February 28, 2013
Copyright: © 2013 Science and Education Publishing. All Rights Reserved.
1. Introduction
In a major way, Boolean algebra differs from mathematics algebra. Boolean constants and variables are allowed to have only two logic values, 0 or 1, giving an output of these two possibilities ^{[1, 2, 3]}.
Boolean Algebra is the algebra for digital signals. Standard rules and laws of this unique Algebra are derived from logic gates characteristics (ANDgate and ORgate).Few characteristics are shown in Figure 1 and Figure 2.
Figure 1. Basic Multiplications for Boolean Algebra (characteristic of ANDgate)
Figure 2. Basic Additions for Boolean Algebra (characteristic of ORgate)
From these basic rules, other rules are being derived as follows:
 (1) 
 (2) 
 (3) 
 (4) 
Beside rules, standard laws also have been set up for Boolean Algebra that generally use the same principles of mathematics algebra ^{[2, 4, 5]}.
A. Commutative Law:
 (5) 
 (6) 
B. Associative Law:
 (7) 
 (8) 
C. Distributive Law:
 (9) 
Those rules and laws will offer you a great help when dealing with a complicated logic equations that need to be worked out.
2. Aims of Study
This paper is trying to derive rules of Boolean Algebra related to a logic equation of an exclusiveor (XOR) gate. There is a hope that this method will be useful for students, lecturers, and anyone that are interested in learning number system, or to whom that are taking subject related to computer or digital system.
3. Research Method
To derive the rules of Boolean Algebra related to an XOR gate, I take the basic point of standard rules and substitutes the standard rules with an XOR gate function
4. Discussion and Results
XORgate is a basic logic gate in digital system that use the combination of AND and OR gates.
Figure 3. Logic Symbol of XOR Gate
The characteristic of XOR gate is given as follows: ^{[1, 2, 3]}
Table 1. XORGate Truth Table
If the number of logic 1s is Odd, the output will be 1, otherwise it will be 0.
Using the characteristic of XOR gate, substitutes the standard rules of Boolean Algebra with this function.If needed, apply Boolean Algebra laws.
 (10) 
 (11) 
 (12) 
 (13) 
 (14) 
 (15) 
 (16) 
 (17) 
 (18) 
So, we got the conclusion and rules for XOR Boolean algebra:
5. Conclusion
Implementing XORgate characteristics into the standard rules and laws of Boolean algebra, will create a specific Boolean algebra that uses XOR function. It gives another way of solving logic equations which are written or given in XOR logic operator.
Acknowledgement
I would like to thank my colleagues and students for the moments and memories while working as a lecturer in my university.
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