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Boolean algebra is a branch of mathematics that deals with logical value operations on binary variables. In boolean algebra, variables can only have one of two potential values: 1 or 0.
- George Boole first introduced the concept of boolean algebra in his book The Mathematical Analysis of Logic.
- Binary algebra or Logical algebra is another name for Boolean algebra.
- Most typically, Boolean variables are provided with the options of "true" or "false".
- Boolean algebra's three major logical operations are conjunction, disjunction, and negation.
- The method is most commonly used in solving problems related to digital circuits or gates.
- The process involves enclosing the elements into binary operations that the postulate can describe.
- Variables can also be interpreted in more complicated ways, as in set theory with Venn Diagrams.
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Key Terms: Boolean Algebra, Conjunction, Disjunction, Negation, Demorgan’s Law, Association, Commutation, Distribution, Boolean Operation, Set Theory
What is Boolean Algebra?
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Boolean algebra uses binary variables to accomplish logical operations. These variables indicate the truth values, which can be either 0 or 1. Conjunction, disjunction, and negation are the three basic boolean operations.
- These operations are conjunction (∧), disjunction (∨) and negation (¬).
- Boolean Algebra is represented by the logical operators AND, OR, and NOT, respectively.
- In addition, these operations are comparable to set theory's intersection, union, and complement of sets.
Boolean Algebra
For example, if we have two variables, A and B, with values of 1 and 0. We must carry out the AND operation. A.B = 1.0 = 0 can be used to represent the boolean expression.
- The boolean statement is A + B = 1 + 0 = 1 if we have to conduct the logical OR operation.
- When we use the NOT function on both input variables, we obtain A = 0 and B = 1.
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Boolean Algebra Expression
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Logic operators such as AND, OR, NOT, and others are used in Boolean algebra expressions. These logical statements have two possible outcomes: true or false.
- In digital circuits and logic gates, the input and output conditions are represented by the numbers "1" and "0."
- For example, we get a boolean expression when we write A OR B.
- Some numerous laws and theorems can be utilised to simplify boolean algebra expressions.
- This will help increase the performance of digital circuits and optimise calculations.
Boolean Algebra Operations
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There are three basic operations in Boolean algebra conjunction, disjunction, and negation. The logical operators that are comparable to these operations are listed below.
Boolean Algebra Operations
AND Operator
The AND operator is similar to the conjunction operator. It is represented by "•" in a boolean expression. The condition for the AND operator states that If and only if all of the input operands are true, this operator returns true. It is also known as a conjunction operator.
OR Operator
The OR operator is the same as the disjunction operator. It is represented by the "+" symbol in a boolean expression. The condition for the OR operator states that If and only if one or more of the input operands is true, the operator returns true. It is also known as a disjunction operator.
NOT Operator
The NOT operator is a logical operator that is similar to negation. If the input variable is false, the NOT operator returns true. In the same way, if the input variable is false, the result is true. An overline on the variable represents this operator. It is also known as the negation operator.
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Boolean Algebra Truth Table
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When the above procedures are expressed in a truth table, we get the following table.
| X | Y | X ∧ Y (Conjunction) | X ∨ Y (Disjunction) |
|---|---|---|---|
| True | True | True | True |
| True | False | False | True |
| False | True | False | True |
| False | False | False | False |
The truth table for negation operation is as follows:
| X | ¬X (Negation) |
|---|---|
| True | False |
| False | True |
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Boolean Algebra Terminologies
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Important terminologies used in Boolean algebra are discussed below.
Boolean Algebra
Boolean algebra is an area of mathematics that deals with logical operations and binary variables.
Literal
A literal can be either a variable or a variable's complement.
Boolean Variable
A boolean variable is a variable or a symbol, usually an alphabet, that expresses logical amounts like 0 or 1.
Boolean Function
Binary variables, logical operators, constants such as 0 and 1, equal to the operator, and the parenthesis symbols make up a boolean function.
Complement
The complement is the inverse of a variable, and it is symbolised by a bar over the variable.
Truth Table
The truth table is a table that lists all of the potential values of logical variables as well as their combinations. The boolean equation can be transformed into a truth table.
- The truth table should have a number of rows equal to 2n, where "n" is the number of variables in the equation.
- For instance, if a boolean equation has two variables, the truth table will have four rows.
- For example: 22=4
Boolean Algebra Rules
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The rules of boolean algebra are listed below.
- Only two values are allowed for the variable.
- Binary ‘1’ denotes HIGH, while Binary ‘0’ denotes LOW.
- An overbar represents the complement of a variable.
- As a result, the complement of variable X is written as X.
- If X = 0 then X=1
- A plus ‘+’ sign between the variables indicates that they are OR operated.
- For example, OR-operated X, Y, and Z are represented as X + Y + Z.
- Writing a dot between two or more variables, such as X.Y.Z., represents logical AND.
- In some cases, such as XYZ, the dot may be omitted.
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Postulates of Boolean Algebra
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A postulate is a statement that is presumed to be true even when it hasn't been proven. In Boolean algebra, postulates are statements that hold rather than rules or theorems. The four potential logical OR and logical AND operations, as well as the rules followed by the NOT operator, are included in these postulates.
- 0 + 0 = 0
- 0 + 1 = 1
- 1 + 0 = 1
- 1 + 1 = 1
- 0 . 0 = 0
- 0 . 1 = 0
- 1 . 0 = 0
- 1 . 1 = 1
- 1' = 0
- 0' = 1
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Laws of Boolean Algebra
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The most common application of boolean algebra is to simplify logic circuits. Using Boolean algebra laws, we can simplify a logical expression and reduce the number of logic gates required in a digital circuit. Important Boolean laws are discussed below.
Associative Law
If more than two variables are OR'd or AND'd, the order in which they are grouped does not matter, according to the associative law. The outcome is always the same.
- X + (Y + Z) = (X + Y) + Z
- X.(Y.Z) = (X.Y). Z
Commutative Law
The commutative law asserts that if the order of the operands (AND or OR) is changed, the result of the boolean equation remains the same.
- X + Y = Y + X
- X.Y = Y.X
Distributive Law
According to the distributive law, if we AND two variables and OR the result with another variable, the result is equal to the AND of the OR of the third variable with each of the first two variables.
- X + Y.Z = (X + Y) (X + Z)
Thus, OR distributes over AND
If two variables are ORed, then the result is ANDed with another variable; the result is equal to the OR of the AND of the third variable with the other two variables.
- X .(Y+Z) = (X.Y) + (X.Z)
Hence, AND distributes over OR.
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Absorption Law
By absorbing like variables, the absorption law connects binary variables and aids in the reduction of difficult formulations. This law applies to four different assertions.
- X + X.Y = X
- X (X + Y) = X
- X + X.Y = X + Y
- X.(X + Y) = X.Y
The following are some more characteristics and identities of boolean algebra.
- X + 1 = 1
- X + 0 = X
- X . 1 = X
- X . 0 = 0
- X + X = X
- X . X = X
- X'' = X
- X + X' = 1
- X.X' = 0
OR Law
In OR law, OR operation is used as mentioned below.
- X + 0 = X
- X + 1 = 1
- X + X = X
- X + X' = 1
AND Law
In AND law, AND operation is used as mentioned below.
- X .0 = 0
- X . 1 = X
- 1. X = X
- X.X' = 0
Inversion Law
The NOT operation is used in this legislation. According to the inversion law, double inversion of a variable results in the original variable.
- X + X'' = 1
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Boolean Algebra Theorem
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De Morgan's theorem is one of the most important theorems in boolean algebra. The AND, OR, and NOT operators are linked in this theorem by two propositions.
De Morgan's First Law
According to De Morgan's First Law.
- (X.Y)' = X'+Y'
The first law asserts that the sum of the individual complements of a variable is equal to the complement of the product of the variables.
The following is the truth table that verifies Demorgan's First Law.
| X | Y | X' | Y' | (X.Y)' | X'+Y' |
|---|---|---|---|---|---|
| 0 | 0 | 1 | 1 | 1 | 1 |
| 0 | 1 | 1 | 0 | 1 | 1 |
| 1 | 0 | 0 | 1 | 1 | 1 |
| 1 | 1 | 0 | 0 | 0 | 0 |
The last two columns in the truth table prove that (X.Y)' = X'+Y'. Hence, Demorgan's First Law is proved.
De Morgan's Second Law
According to De Morgan's Second Law, (X+Y)' = X'. Y'.
- The complement of the sum of variables is equal to the product of their complements of a variable, according to the second law.
- The truth table below shows the proof for De Morgan's second law.
| X | Y | X' | Y' | (X+Y)' | X'Y' |
|---|---|---|---|---|---|
| 0 | 0 | 1 | 1 | 1 | 1 |
| 0 | 1 | 1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 1 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
The last two columns in the truth table prove that (X+Y)' = X'. Y'. Hence, Demorgan’s second law is proved.
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Logic Gates
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A logic gate is a component in digital circuits that serves as a building block. They carry out basic logical operations that are essential in digital circuitry.
- Logic gates in a circuit make judgments depending on a mix of digital signals from their inputs.
- There are two inputs and only one output on most logic gates.
- Boolean algebra is used to create these logic gates listed below with their truth tables.
Logic Gates
AND Gate
If both X and Y are true then only Z will be true. The expression for AND gate is Z = X.Y
| X | Y | Z = X.Y |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
OR Gate
If any of X or Y is true, Z will be true. The expression for OR gate is Z = X + Y
| X | Y | Z = X + Y |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 1 |
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NOT Gate
The output Z is true only if X is false, it is also known as invertible or inverter gate. The expression for NOT gate is Z = X'
| X | Z = X' |
|---|---|
| 1 | 0 |
| 0 | 1 |
NAND Gate
The NAND gate is the outcome of the NOT - AND operation.
- Z will NOT be true if X and Y both are true, and Z = (X.Y)' symbolises the boolean equation.
| X | Y | Z = (X.Y)' |
|---|---|---|
| 0 | 0 | 1 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
NOR Gate
The NOR gate is the outcome of the NOT - OR operation.
- Z is true if X and Y are not true, and Z = (X + Y)' symbolises the boolean equation.
| X | Y | Z = (X + Y)' |
|---|---|---|
| 0 | 0 | 1 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 0 |
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Things to Remember
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- Boolean algebra is a branch of mathematics that deals with logical operations and binary variables.
- A bar across a variable represents the complement, which is the inverse of the variable.
- Boolean algebra has three basic operations: conjunction, disjunction, and negation.
- These logical propositions can only have one of two outcomes: true or untrue.
- A logic gate is a component that serves as a building block in digital circuits.
- They perform the fundamental logical processes required in digital electronics.
- Associative law, commutative law, distributive law, absorption law, OR law, AND law, and inversion law are all important Boolean laws.
Sample Questions
Ques. What is Boolean Algebra? [3 marks]
Ans. Boolean algebra is a branch of mathematics that deals with logical value operations and includes binary variables. Boolean algebra may be traced back to a work written by mathematician George Boole in 1854.
- Boolean algebra is distinguished by the fact that it is limited to the study of binary variables.
- The most common values for Boolean variables are 1 ("true") and 0 ("false").
- Variables can also be interpreted in more complicated ways, as in set theory.
- Binary algebra is another name for Boolean algebra.
Ques. How is Boolean Algebra different from Elementary Algebra? [5 marks]
Ans. Boolean algebra, or boolean equation, evaluates solely binary in terms of truth, while elementary algebra works with polynomials and numeric values. In addition, Boolean algebra executes conjunction, disjunction, and negation, in contrast to regular algebra's operations of addition, subtraction, division, and multiplication.
- The variables in Boolean algebra can only have logical values, hence the sum of two identical variables equals the variable itself.
- Similarly, the variable itself is the product of two identical variables.
- In contrast to Boolean rules, linear algebra assigns a numerical value to the variables.
- As a result, the answer is found by adding the coefficients of two identical variables.
- Similarly, the product of two identical variables is the product of the coefficient and the square of the variable in multiplication.
- There are no fractional or negative numbers in Boolean algebra rules.
- It merely assigns a binary value of 0 or 1, corresponding to false and true, respectively.
- Integers, fractions, rational numbers, and other concepts are covered in elementary algebra.
Ques. What are the applications of Boolean Algebra? [3 marks]
Ans. In finance, Boolean algebra is utilised to describe market analysis as binomial numbers. In electronics and computer programming, Boolean expressions for logic gates play a significant role.
- It is utilised in digital circuitry, which is used in personal computers, CD players, and other electronic devices.
- Boolean operations are also employed in search engines to display relevant search results.
- It is used in the study of statistics in mathematics.
Ques. What are the three main Boolean Operations? [3 marks]
Ans. Three main Boolean Operations are Conjunction, Disjunction and Negation.
- AND operator: It is similar to the conjunction operator in that it accepts two values. In a boolean statement, the AND operator is represented by "•." This operator returns true if and only if all of the input operands are true.
- OR operator: In a boolean expression, the OR operator is represented by the "+" sign. The operator returns true if and only if one or more of the input operands are true.
- NOT operator: It is a logical operator that works similarly to negation. The NOT operator returns true if the given variable is false. Similarly, if the input variable is false, the outcome is also false. An overline on the variable represents this operation.
Ques. List main Boolean Laws. [1 mark]
Ans. Important Boolean laws are
- Associative Law
- Commutative Law
- Distributive Law
- Absorption Law
- OR Law
- AND Law
- Inversion Law
Ques. Evaluate the output Truth Table for X(Y+Z). [3 marks]
Ans. The truth table for X(Y+Z) output is given in the table below.
| X | Y | Z | Y + Z | X(Y + Z) |
|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 |
| 0 | 1 | 0 | 1 | 0 |
| 0 | 1 | 1 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 1 | 1 | 1 |
| 1 | 1 | 0 | 1 | 1 |
| 1 | 1 | 1 | 1 | 1 |
Ques. Evaluate the output Truth Table for Z = X'.Y [3 marks]
Ans. The truth table for X'.Y output is given in the table below.
| X | Y | X' | X'.Y |
|---|---|---|---|
| 0 | 0 | 1 | 0 |
| 0 | 1 | 1 | 1 |
| 1 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 |
Ques. Define Logic Gates. [2 marks]
Ans. A logic gate is a fundamental component of every digital circuit. These logic gates must decide whether to combine numerous inputs and produce an output based on some logical procedure. Logic gates use boolean algebra to execute logical processes in various fields.
Ques. What is a NAND gate? [2 marks]
Ans. A digital combination logic circuit with two inputs that performs the logical inverse of an AND gate is known as a two-input NAND gate. While an AND gate only produces a logical "1" if both inputs are logical "1," a NAND gate emits a logical "0" when the same inputs are used.
Ques. What is a NOR gate? [2 marks]
Ans. A NOR gate is an OR gate followed by a NOT gate in a cascade. In other terms, a NOR gate is a gate that produces a strong output signal only when the input values are low.
Ques. Draw a truth table for A(B’ + D). [3 marks]
Ans. Given expression A(B’ + D).
| A | B | B’ | D | B’ + D | A( B’ + D) |
|---|---|---|---|---|---|
| 0 | 0 | 1 | 0 | 1 | 0 |
| 0 | 0 | 1 | 1 | 1 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 1 | 1 | 0 |
| 1 | 0 | 1 | 0 | 1 | 1 |
| 1 | 0 | 1 | 1 | 1 | 1 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 1 | 1 | 1 |
Ques. Evaluate the output Truth Table for X + Y+Z. [3 marks]
Ans. The truth table for X(Y+Z) output is given in the table below.
| X | Y | Z | Y + Z | X + Y + Z |
|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 1 |
| 0 | 1 | 0 | 1 | 1 |
| 0 | 1 | 1 | 1 | 1 |
| 1 | 0 | 0 | 0 | 1 |
| 1 | 0 | 1 | 1 | 1 |
| 1 | 1 | 0 | 1 | 1 |
| 1 | 1 | 1 | 1 | 1 |
Ques. Evaluate the output Truth Table for X(Y’+Z’). [3 marks]
Ans. The truth table for X(Y+Z) output is given in the table below.
| X | Y | Y’ | Z | Z’ | Y’ + Z’ | X(Y’ + Z’) |
|---|---|---|---|---|---|---|
| 0 | 0 | 1 | 0 | 1 | 1 | 0 |
| 0 | 0 | 1 | 1 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 | 1 | 1 | 0 |
| 0 | 1 | 0 | 1 | 0 | 1 | 0 |
| 1 | 0 | 1 | 0 | 1 | 1 | 1 |
| 1 | 0 | 1 | 1 | 0 | 1 | 1 |
| 1 | 1 | 0 | 0 | 1 | 1 | 1 |
| 1 | 1 | 0 | 1 | 0 | 1 | 1 |
Ques. Evaluate the output Truth Table for X + YZ. [3 marks]
Ans. The truth table for X(Y+Z) output is given in the table below.
| X | Y | Z | YZ | X + YZ |
|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 1 | 1 | 1 | 1 |
| 1 | 0 | 0 | 0 | 1 |
| 1 | 0 | 1 | 0 | 1 |
| 1 | 1 | 0 | 0 | 1 |
| 1 | 1 | 1 | 1 | 1 |
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