# Boolean Logic - Explained

What is Boolean Logic?

# What is Boolean Logic?

In mathematical concepts and mathematical logic, the term Boolean algebra refers to the branch of algebra that deals with the true value of variables. These values are based on determining whether a number is true or false and usually assigned 1 and 0 respectively. Rather than elementary algebra, where the values are variable numbers, and the prime operations are multiplication and addition, the key operations of Boolean algebra based on conjunctions and disjunction denoted by , and , respectively and the negation does not have denotation as . Therefore, it is a formula used to describe logical relations in a similar way the elementary algebra describes numeric relations.

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## How does Boolean Logic Work?

The root of Boolean algebra can be traced to 1847 when George Boole introduced his first book The Mathematical Analysis of Logic (1847) and advanced it fully in his An Investigation of the Laws of Thought (1854). Huntington presents that the term "Boolean algebra" was proposed first by Sheffer in 1913 though Charles Sanders Peirce introduced the title "A Boolean Algebra with One Constant in 1880 in the first chapter of his book "The Simplest Mathematics." The concept of Boolean algebra has considerably helped in the development of digital electronic by enhancing computer programming languages. Besides, it is also applied in set statistics and theories While the elementary algebraic expressions mainly concern with numbers, Boolean algebraic expression concerns with truth value false and true. The Boolean algebraic numbers are represented in either binary digits or bits which are either 0 or 1. These numbers do not have characteristics of integers 0 and 1 whereby 1+1 =2. However, they may be identified with elements of the two-element field GF (2). In other words, the integer arithmetic modulo 2 which provides that 1+1 = 0. The multiplication and addition paly the Boolean role of XOR (exclusive=or) and AND (conjunction) respectively. On the other hand, the disjunction xy (inclusive-or) is described as x + y - xy. Moreover, Boolean algebra operates with functions that have values lying between {0, 1}. The function uses a sequence of bits to generate values. One of the best examples is a subset of set E: to a subset F of E. This means that the indicator of the function will take the value 1 of F and 0 outside F. In addition, the most general example is the Boolean algebraic elements that all the preceding remains the instance thereof. As in the case of elementary algebra, the explicitly equational part of the theory can be formed without considering the values attached to the variables

## Basic operations

The Boolean algebraic operation is demonstrated below:

• AND (conjunction), designated xy (sometimes x AND y or Kxy), contents xy = 1 if x = y = 1 and xy = 0 otherwise.
• OR (disjunction), designated xy (sometimes x OR y or Axy), contents xy = 0 if x = y = 0 and xy = 1 otherwise.
• NOT (negation), designated x (sometimes NOT x, Nx or !x), contents x = 0 if x = 1 and x = 1 if x = 0.

## Secondary operations

The above operation of Boolean algebra is considered basic operation because they can form the basis for other Boolean operations. These operations can be built up by composing the ways of combining or compounding operations. The first operation known as the material implication is presented by, x y, or Cxy. In this case, if x variable is true, then the value of x y is considered to be that of y. On the other hand, if x variable is false, the value of y can be ignored; however, the operation must reveal the value of Boolean that has only two choices available. Therefore, by definition, x y is true when x is false. (Relevance logic proposed the definition by considering the implication with a false variable as a different concept other than either true or false. The second operation known as exclusive (XOR) is denoted by, x y, or Jxy, is to distinguish it from inclusive disjunction. This operation excludes the likelihood of both x and y. this operation id defined in the arithmetic as addition mod 2 that presents that 1+1 =0 The third operation, is known as the complement of exclusive or, is Boolean equality or equivalence: x y, or Exy, is considered to be true only when x and y have similar value. Therefore, the complement x y can be understood as x y, being true only when x and y are different. The counterpart of Equivalence in arithmetic mod 2 is x + y + 1. Given two operands, that have two possible values, 22= 4 will be generated as possible combination inputs since the output of each value can have two possible values the sum of 24=16 possible binary Boolean operation.