Introduction
Boolean algebra was formulated by George Boole who was a self-educated English mathematician between the periods 1815 to 1864. It basically involves the triumphant application of algebraic techniques to logic. The algebraic laws developed by Boole are fundamentally similar to those of binary arithmetic where essential symbols are inferred as assuming only the digit values 0 and 1 commonly referred to as either true or false. Boole considered these basic symbols capable of being combined through definite functions: for instance, multiplication related to the union of elements or sets of elements, subtraction referring to differences in elements or sets of elements, addition referring disjoint combination. The binary operator utilizes a couple of Boolean inputs to generate a sole output.
Main body
The Boolean algebra can be identified as set B consisting of elements “a, b” with the following features:
- The set B consists of two binary processes,^̂̌• (logical AND) and ̌ + (logical OR) that suit the idempotent law that states a• a=a + a=a, the commutative law that states a • b=a • b and a + b=a + b. thirdly, the associative law of the Boolean algebra states that a•( b •c )=(a •b)• c and a + (b + c)=(a+ b) + c.
- There are other laws that combine both the binary operators. These are: a • (a + b) =a+ (b • a) =a, that utilizes the absorption law of the algebra.
- Boolean algebra is mutually distributive if it satisfies the following law: a• (b + c) = (a • b) + (ac) and a+ (b • c) = (a + b) • (a + c).
- The set B is conceived to consist of two common limits φ and that satisfy the following laws: φ • a=φ, φ + a=a and I • a=a, I + a= I. In addition, the set contains the unary operator a’ meant for harmonization and lies under the following laws: a • a’= φ and a + a’=I.
- The Boolean structure is closed for the binary operators AND, OR and NOT if and only if for every set of Boolean inputs, it gives a single Boolean output. For instance the Boolean operator AND is closed mainly because for every binary operator that is a Boolean outcome.
- Additionally, if “a” and b belong to the set B then it implies that a • b and a + b also belong the set B. in presence of the element Z (0) then the following rule must be satisfied a + Z= a and the presence of the element U (unity or one) then it implies that a • U = a for each element a.
- De Morgan further developed the Boolean algebra and came up with his own laws to include more sets of elements with more elements.
The Boolean laws of algebra can easily be proved to apply successive by use of the truth tables or replacing the elements with either values 0 and 1 as the algebra is bound to only this two values. For instance, the commutative law (a • b=a • b) with binary operator AND can be verified letting a=0 and b=1, then replacing the values in the formula 0*1=0*1=0, hence the law is true. The method becomes more complex as more variables are included in the formula and the binary operators interchanged in the same formula. However, the method is easy to apply and understand once you capture the formula as it involves replacing the letters with numbers and the operators with the real mathematical signs such as addition, multiplication etc. On the other hand, the truth tables can be applied to verify the laws. For instance, consider table 1 that uses the binary operator AND assuming that elements “a” and b assume the values 0 and 1 respectively.
Table 1
The above table is normally used as it is easy to understand and convenient to draw. In addition, it can be applied where more than two variables are present to verify the valid of the Boolean laws of his theorem. However, there are more complex tables that combine all variables and the binary operators. For instance, the truth tables used to verify the de Morgan’s laws of his theorem are more complex.
Conclusion
The Boolean algebra relies on sets of elements and their corresponding elements to verify the basics of the theorem; the theorem is confined to the values 0 and 1. This restriction of the algebra makes it different from the usual arithmetic algebra however all other basics are the same. The Boolean algebra has some underlying rules for it to be valid and considered correct. These rules dictate the proper application of the algebra basics and assist in understanding the theorem better and easily. The laws of the algebra have been verified to be true through the use of the formula method or the truth table method. More so, the Boolean algebra theorem has been applied in computer design, probability theory among others hence relevant in the outer universe.
References
Bell, E. T. Men of Mathematics. New York: Simon and Schuster publishers, 1986.
Boole George, an Investigation of the Laws of Thought, Cambridge: Cambridge university press, 2003.