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# Step By Step Boolean Algebra Simplification Calculator

## How To Use The Boolean Calculator

Example Problems Boolean Expression Simplification

Follow the 2 steps guide to find the truth table using the boolean calculator.

• Enter the Expression.
• Click “Parse
• Take help from sample expressions in the input box or have a look at the boolean functions in the content to understand the mathematical operations used in expressions.

Now we are solving above expression using boolean theorems:

 References:

## Boolean Algebra Laws And Rules

There are three laws of Boolean Algebra that are the same as ordinary algebra.

• The Commutative Lawaddition A + B = B + A multiplication AB = BA
• The Associative Lawaddition A + = + C multiplication A = C
• The Distributive Law – The distributive law is the factoring law. A common variable can be factored from an expression just as in ordinary algebra.A = AB + AC
• = AC + AD + BC + BD Remeber FOIL?

## Simplification Of Combinational Logic Circuits Using Boolean Algebra

• Complex combinational logic circuits must be reduced without changing the function of the circuit.
• Reduction of a logic circuit means the same logic function with fewer gates and/or inputs.
• The first step to reducing a logic circuit is to write the Boolean Equation for the logic function.
• The next step is to apply as many rules and laws as possible in order to decrease the number of terms and variables in the expression.
• To apply the rules of Boolean Algebra it is often helpful to first remove any parentheses or brackets.
• After removal of the parentheses, common terms or factors may be removed leaving terms that can be reduced by the rules of Boolean Algebra.
• The final step is to draw the logic diagram for the reduced Boolean Expression.

Perform FOIL

AA = A

Find a like term and pull it out. . Make sure you leave the BC alone at the end.

Anything ORed with a 1 is a 1 .

Anthing ANDed with a 1 is itself

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## How Does A Boolean Algebra Calculator Work

A Boolean Algebra Calculator works by first breaking down a Boolean Algebraic expression into its constituent logical functions, and then calculating each instance according to the rules of precedence.

The rules of precedence in Boolean algebra tend to work very much like the ones in mathematical algebra. A numerical operator applied on a set of parentheses is applied to everything present within the parenthesis. The same is the case with Boolean algebra where a logical gate is applied to every entry present within the parenthesis.

This is how a Boolean algebraic equation is simplified and then solved.

## What Is Boolean Algebra Mathematics has different branches e.g algebra, geometry e.t.c. These branches are further divided into sub-branches. Boolean algebra is one such sub-branch of algebra.

It has two binary values including true and false that are represented by 0 and 1. Where 1 is considered as true and 0 is considered as false.

Boolean expressions are simplified to build easy logic circuits.

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## How To Use Boolean Simplification For Electromechanical Relay Circuits

Electromechanical relay circuits, typically being slower, consuming more electrical power to operate, costing more, and having a shorter average life than their semiconductor counterparts, benefit dramatically from Boolean simplification. Lets consider an example circuit:

As before, our first step in reducing this circuit to its simplest form must be to develop a Boolean expression from the schematic.

The easiest way Ive found to do this is to follow the same steps Id normally follow to reduce a series-parallel resistor network to a single, total resistance.

For example, examine the following resistor network with its resistors arranged in the same connection pattern as the relay contacts in the former circuit, and corresponding total resistance formula:

In the above figure, a long dash symbol is used to represent the series connection of resistors.

Remember that parallel contacts are equivalent to Boolean addition, while series contacts are equivalent to Boolean multiplication.

Write a Boolean expression for this relay contact circuit, following the same order of precedence that you would follow in reducing a series-parallel resistor network to a total resistance.

It may be helpful to write a Boolean sub-expression to the left of each ladder rung, to help organize your expression-writing:

Now that we have a Boolean expression to work with, we need to apply the rules of Boolean algebra to reduce the expression to its simplest form :

REVIEW:

## How To Find Boolean Algebra

The Boolean variables are represented as binary numbers to represent truths i.e., 1 = true and 0 = false. Elementary algebra deals with numerical operations whereas Boolean algebra deals with logistical operations. There are different types of logic gates i.e., AND, OR, and NOR gates.

Conjunction or AND gate: Consider the statement p and q, denoted pq

• Rule1: If ‘p’ and ‘q’ both statements are True then p and q is also a True statement.
• Rule2: If ‘p’ is False while ‘q’ is True then p and q is False. For p and q to be true, we would need Both statements to be True. Since one is false, p and q is False.
• Rule3: If ‘p’ is True while ‘q’ is False then p and q is False. For p and q to be true, we would need Both statements to be True. Since one is false, p and q is False.
• Rule4: If both the statements are False then p and q is False.

Disjunction or OR gate: Consider the statement p OR q

• Rule1: If both the statements are True then “p or q is also a True statement.
• Rule2: If ‘p’ is False while ‘q’ is True then p or q is True. Since one is True.
• Rule3: If ‘p’ is True while ‘q’ is False then p or q is True. Since one is True.
• Rule4: If both the statements are False then “p or q is False.

XOR gate: Consider the statement p XOR q

NOR gate: Consider the statement p NOR q

Negation or NOT gate: Negation is the statement represented by ¬p, and so it would have the opposite Truth value of p. If p is True, then ¬p is False. If p is False, then ¬p is True.

## How To Write A Boolean Expression To Simplify Circuits

Our first step in simplification must be to write a Boolean expression for this circuit.

This task is easily performed step by step if we start by writing sub-expressions at the output of each gate, corresponding to the respective input signals for each gate.

Remember that OR gates are equivalent to Boolean addition, while AND gates are equivalent to Boolean multiplication.

For example, Ill write sub-expressions at the outputs of the first three gates:

. . . then another sub-expression for the next gate:

Finally, the output is seen to be equal to the expression AB + BC:

Now that we have a Boolean expression to work with, we need to apply the rules of Boolean algebra to reduce the expression to its simplest form :

The final expression, B, is much simpler than the original, yet performs the same function.

If you would like to verify this, you may generate a truth table for both expressions and determine Qs status for all eight logic-state combinations of A, B, and C, for both circuits. The two truth tables should be identical.

## What Is A Boolean Algebra Calculator

Boolean Algebra 1 The Laws of Boolean Algebra

A Boolean Algebra Calculatoris a calculator which you can use to solve your Boolean Algebraic expressions online.

This calculator works in your browser via the internet and solves the given problem for you. The calculator is designed to solve Boolean expressions denoted in the correct format.

The Boolean Algebra Calculator, therefore, receives an expression with logic gates correlating the quantities given. These logic gates here are similar to numerical operators in standard algebraic equations.

You can enter your problems in the input box available, where the logic gates have to be typed into the system like \$AND\$, \$OR\$, etc.

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## What Is 4 Variables Karnaugh’s Map

4 Variables Karnaugh’s Map often known as 4 variables K-Map. It’s an alternate method to solve or minimize the Boolean expressions based on AND, OR & NOT gates logical expressions or truth tables. The four variables A, B, C & D are the binary numbers which are used to address the min-term SOP of the Boolean expressions. The gray code conversion method is used to address the cells of KMAP table.

The min-term SOP is often denoted by either ABCD, 1s & 0s or . For example, the Boolean expression y = represents the place values of the respective cells which has the higher values . The y = can also be represented by y = or y = A is the most significant bit and B is the least significant bit . Each variable A, B, C & D equals to value 1. Similarly, each inverted variable A, B, C& D equals to 0. Any 4 combinations of A, B, C, D, A, B, C& D represents the place values of 0 to 15 to address the cells of table in KMAP solver.

Users may refer the below details to learn more about 4 variables Karnaugh’s map or use this online calculator to solve the SOP or generate the complete work for minimum SOP for 4 variables A, B, C & D.

## Boolean Algebra Calculator + Online Solver With Free Steps

A Boolean Algebra Calculator is used to calculate Boolean logic and solve simple as well as complex Boolean Algebraic problems.

This calculator can solve the different properties of Boolean Algebra, catering for commutative, associative, etc, which makes it best for solving complex Boolean Algebraic expressions.

The Boolean Logic here corresponds to the binary logical values which are used to represent mathematical results where the inputs vary from one binary state to another to generate an output response in the system.

## Generating Schematic Diagrams From Boolean Expressions

Now, we must generate a schematic diagram from this Boolean expression.

To do this, evaluate the expression, following proper mathematical order of operations , and draw gates for each step.

Remember again that OR gates are equivalent to Boolean addition, while AND gates are equivalent to Boolean multiplication.

In this case, we would begin with the sub-expression A + C, which is an OR gate:

The next step in evaluating the expression B is to multiply the signal B by the output of the previous gate :

Obviously, this circuit is much simpler than the original, having only two logic gates instead of five.

Such component reduction results in higher operating speed , less power consumption, less cost, and greater reliability.

## The Universal Capability Of Nand And Nor Gates • NAND and NOR gates are universal logic gates.
• The AND, Or, Nor and Inverter functions can all be performed using only NAND gates.
• The AND, OR, NAND and Inverter functions can all be performed using only NOR gates.
• An inverter can be made from a NAND or a NOR by connecting all inputs of the gate together.
• If the output of a NAND gate is inverted, it becomes an AND function.
• If the output of a NOR gate is inverted, it becomes an OR function.
• If the inputs to a NAND gate are inverted, the gate becomes an OR function.
• If the inputs to a NOR gate are inverted, the gate becomes an AND function.
• When NAND gates are used to make the OR function and the output is inverted, the function becomes NOR.
• When NOR gates are used to jake the AND function and the output is inverted, the function becomes NAND.

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## How To Use The Boolean Algebra Calculator

To use the Boolean Algebra Calculator properly, a set of instructions must be followed. First, you must have a Boolean Algebraic expression to solve. In this expression, the gates are to be expressed as \$AND\$, \$OR\$, etc., therefore, no symbols are to be used.

The use of parenthesis in the proper fashion is very important. The lack of parenthesis can get the calculator confused and cause problems.

## How To Solve 4 Variables Kmap

Users may refer the below rules & step by step procedure to learn how to find the minimum sum of products for the Boolean expression using 4 variables A, B, C & D. When you try yourself solving the min-term SOP of for 3 variables, Users can use this online Karnaugh’s map solver for 4 variables to verify the results of manual calculations.

step 1 Addressing the cells of KMap tableWhen using KMAP solver, generally users should be careful while placing the min-terms. Because, the addressing of min-terms in KMAP table is bit different. The order of the cells are based on the Gray-code method. Refer the below table & information gives the idea of how to group the KMAP cells together. For four variables, the location of the the cells of KMAP table as followsIn Binary Form

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## Application Of Boolean Algebra

Combinational Logic Circuit Design comprises the following steps

• From the design specification, obtain the truth table
• From the truth table, derive the Sum of Products Boolean Expression.
• Use Boolean Algebra to simplify the boolean expression. The simpler the boolean expression, the less logic gates will be used.
• Use logic gates to implement the simplified Boolean Expression.
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## Frequently Asked Questions On Boolean Algebra

Simplification of Boolean Expression using Boolean Algebra Rules | Important Question 2

1. What is meant by Boolean Algebra?

Boolean algebra is a branch of mathematics that deals with the operations on logical values. It returns only two values i.e true or false or represented by 0 and 1.

2. What are the operations used in the boolean algebra?

The various basic operations used in the boolean algebra are Conjunction , Disjunction, and Negotiation .

3. How do you calculate the Boolean Algebra Expression using a calculator?

Enter a valid boolean expression and hit on the calculate button to get your answer quickly.

4. What are the 7 logic gates?

There are seven basic logic gates. They are AND, OR, XOR, NOT, NAND, and XNOR.

5. What is the other name of Boolean Algebra?

Boolean Algebra is used to simplify and analyze the digital circuits. It has only the binary numbers i.e. 0 and 1. It is also called Binary Algebra or logical Algebra.