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Boolean Algebra, Logical Equations, and Logical Functions

Posted in Digital Circuit

In the previous chapter on binary arithmetic, we discussed logical operations such as AND, OR, and NOT. However, when describing things, relying solely on data calculations is not enough. Just as in decimal arithmetic, in binary arithmetic, there should be algebraic expressions, logical equations, and logical functions.

  • Boolean algebra studies general logical operation rules,
  • Logical equations solve input-output relationships that meet specific requirements, and
  • Logical functions study the laws of output changes with input from a broader range, revealing the laws of things development.

Based on logical functions, we can broadly describe the characteristics, states, and behaviors of things. Therefore, this chapter introduces the concepts of Boolean algebra, logical equations, and logical functions, which can more effectively solve logical relationships between things, as well as reasoning, judgment, and behavior.

Boolean Algrbra, Logical Algrbra
Boolean Algrbra, Logical Algrbra

1. Brief Introduction of  Boolean Algebra

Logic algebra (or algebra of logic) is a mathematical method used to describe the logical relationships between objective things. It was introduced by British scientist George Boole in the mid-19th century, hence it is also called Boolean algebra. Boolean algebra has a complete set of rules for operations, including axioms, theorems, and laws. It is widely used in the transformation, analysis, simplification, and design of switch and digital logic circuits, and is therefore also known as switch algebra.

With the development of digital technology, Boolean algebra has become the basic tool and theoretical foundation for analyzing and designing logic circuits. Logic equations established using logic algebra provide models for describing the characteristics of things, relationships between things, and motion laws, which is what we commonly refer to as modeling. Solving equations involves finding the necessary solutions based on logical modeling.

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Boolean algebra and logic equations are similar to the algebra and equations in general mathematics. After fully understanding the characteristics, logic, and development laws of things, specific letters such as X, Y, A, B, C, etc. are used to replace actual numbers, logical or temporal expressions are written, logic equations are established, logic expressions are simplified, and equations are solved to obtain corresponding output conditions that satisfy specific output requirements.

Logic functions describe the variation law of output with input, establish the input-output relationship, simplify the logic expressions in the function, and finally realize the function in the chip by successful RTL engineering.

There are usually two ways to express logic functions:

  • algebraic expressions
  • truth tables.

The algebraic method describes the relationship between logic function input and output through algebra. The truth table describes the input-output relationship by listing (or enumerating) all input and output corresponding values.

For example, the exclusive OR logic function can be described by equation :

Y = A(~B)+(~A)B

which shows the relationship between the input variables A and B and the output Y.

This function can also be represented by a truth table, as shown in Table 1:

Table 1, Exclusive OR function

Input Output
A B Y
0 0 0
0 1 1
1 0 1
1 1 0

2. Research Methods of Boolean Algebra (Logic Algebra)

  1. Symbolic method: It is the main method of Logic Algebra research. It uses symbolic logic to represent logical expressions and logical operations. In this method, logical propositions are represented by symbols, and logical operations are represented by symbols as well. This method is based on the fact that logic operations have specific properties that can be represented by symbols.
  2. Algebraic method: This method is based on the study of algebraic structures. It mainly uses algebraic structures to study logic operations and logical relationships. The algebraic method is mainly used to study the properties of logic operations and the laws of logic operations.
  3. Logical method: The logical method is mainly based on the principles and laws of logic to study logic operations and logical relationships. It mainly uses the method of logical deduction to study the properties of logic operations and the laws of logic operations.
  4. Graphical method: The graphical method is mainly used to visually represent logic operations and logical relationships. It mainly uses logic diagrams to represent logic operations and logical relationships. The graphical method is mainly used for practical applications such as circuit design and digital logic design.
  5. Other methods: In addition to the above methods, there are other methods of studying Logic Algebra, such as the method of semantic analysis, the method of formal proof, and the method of model theory. These methods are used to study the semantics of logic operations, the validity of logical propositions, and the models of logical systems, respectively.
  6. Truth Table: a table used in logic to represent the truth values of logical propositions, such as statements and predicates, for all possible input values. In the table, each row represents a possible combination of input values, and the columns represent the resulting truth values. Truth tables are commonly used in Boolean algebra, propositional logic, and predicate logic. They are a simple and effective tool for analyzing and evaluating logical expressions and are used to determine whether a given logical expression is true or false for all possible input values.
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3. The Research Content of Boolean Algebra (Logic Algebra)

3.1  Logic Function Analysis

  • Analyzing the overall logic function, given the logic function expression or truth table, to analyze the logic function and provide a description of the input-output relationship. For example, given the expression Y = A(~B)+(~A)B or truth table (Table 1), through analysis, we can conclude that only when A and B are different, Y is 1. The input and output have an XOR relationship or a one-bit addition relationship.
  • For specific requirements, establish equations and solve them. For example, according to the function Y = A(~B)+(~A)B, when Y = 1, we can solve for the values of A and B, and we can get (A,B) equals (0,1) or (1,0) to satisfy the requirements.

3.2 Logic Function Synthesis

Logic function synthesis establish the corresponding relationship between logic functions based on requirements or development laws of objects, events, states, etc.

  • Creating a truth table
  • Writing logic function expressions
  • Simplifying the expressions
  • Drawing logic circuits or RTL designs.

Note:

Both “Logic algebra” and “Boolean Algebra” are all OK to describe Logic algebra, but in the field of computer science, “Boolean Algebra” is typically used to describe logic algebra, as this term is more common and specifically refers to the logic algebra proposed by George Boole. However, “Logic algebra” is also a general term in some fields, such as philosophy and mathematics, so the specific term used depends on the context and the application.

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