Boolean algebra Simplification, examples, and solutions

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 treatise written by mathematician George Boole in 1854.

Boolean Algebra

Boolean algebra is distinguished by the fact that it is limited to the study of binary variables. The most typical presentation of Boolean variables is with the alternative values of 1 and 0. (“true”) or 0 (“false”). Variables can also have more complex interpretations, such as in set theory. Boolean algebra is also known as binary algebra.

Key TAKEAWAYS IMPORTANT

  • The field of mathematics known as Boolean algebra deals with operations on logical values with binary variables.
  • To represent truths, Boolean variables are represented as binary numbers: 1 = true and 0 = false.
  • Boolean algebra deals with logistical activities, while elementary algebra deals with numerical operations.
  • In contrast to addition, subtraction, multiplication, and division, Boolean algebra use conjunction, disjunction, and negation.
  • Computer programming languages are the most common current use of Boolean algebra.
  • Binomial options pricing models in finance employ Boolean algebra to assist determine when an option should be exercised.

What is Boolean Algebra?

The study of binary variables is what sets Boolean algebra apart. The most common values for Boolean variables are 1 (“true”) and 0 (“false”) (“false”). Variables can also be interpreted in more complicated ways, as in set theory. Binary algebra is another name for Boolean algebra.

Boolean Algebra Example

Create a Truth Table for the logical functions at points C, D, and Q in the circuit below, and find a single logic gate that can replace the entire circuit.

According to our initial observations, the circuit consists of a two-input NAND gate, a two-input EX-OR gate, and a two-input EX-NOR gate at the output.

Because the circuit only has two inputs, A and B, there can only be four potential input combinations ( 22 ), which are 0-0, 0-1, 1-0, and 1-1. The following truth table for the entire logic circuit can be obtained by tabulating the logical functions from each gate.

input output
A B C D Q
0 0 1 0 0
10 1 1 1 1
1 0 1 1 1
1 1 0 0 1

Column C in the truth table above indicates the NAND gate’s output function, whereas column D represents the Ex-OR gate’s output function. Both of these output expressions are then used as inputs for the Ex-NOR gate at the output.

The truth table shows that when either of the two inputs A or B is at logic 1, an output at Q is present. An OR Gate is the only truth table that meets this requirement. As a result, the entire circuit above can be replaced by a single 2-input OR Gate.

Fundamental Logic Gates

Logic gates are the fundamental components of any digital system. It’s an electrical circuit with only one output and one or more inputs. A specific logic governs the relationship between the input and the output. AND gate, OR gate, NOT gate, and so on are examples of logic gates.

The 7 types of logic gates [logic function, logic symbol, truth table, boolean expression]

AND gate OR gate, XOR gate, NAND gate, NOR gate, XNOR gate, and NOT gate are the seven types of basic logic gates. The logic gate function is depicted using the truth table. Except for the NOT gate, which has only one input, all logic gates have two inputs.

Boolean Algebra/ the 7 types of logic gates

AND gates

The AND gate is an electronic circuit that gives a high output (1) only if all its inputs are high.  A dot (.) is used to show the AND operation i.e. A.B.  Bear in mind that this dot is sometimes omitted i.e. AB

 AND gates/ Boolean Algebra

OR gates

The OR gate is an electronic circuit that gives a high output (1) if one or more of its inputs are high.  A plus (+) is used to show the OR operation. 

OR gates /Boolean Algebra

NOT gates

The NOT gate is a type of electrical circuit that outputs an inverted version of the input. An inverter is another name for it. The inverted output is known as NOT if the input variable is A. As shown at the outputs, this is also displayed as A’, or A with a bar over the top. The diagrams below demonstrate two different ways to arrange the NAND logic gate to form a NOT gate. It’s also possible.

NOT gates/ Boolean Algebra

Universal logic Gate

NAND

This is a NOT-AND gate, which is a combination of an AND gate and a NOT gate. If any of the inputs is low, the outputs of all NAND gates are high. An AND gate with a little circle on the output is the symbol. Inversion is represented by the little circle.

NAND gates/ Boolean Algebra

NOR

This is a NOT-OR gate, which is a combination of an OR gate and a NOT gate. If any of the inputs is high, the outputs of all NOR gates are low.

NOR gates/ Boolean Algebra

An OR gate with a little circle on the output is the symbol. Inversion is represented by the little circle.

Exclusive logic gates

XOR

The ‘Exclusive-OR’ gate is a circuit that produces a high output if either of its two inputs is high, but not both. The EOR operation is shown by an encircled plus symbol ().

XOR gates /Boolean Algebra

XNOR

The ‘Exclusive-NOR’ gate circuit operates in the opposite direction from the EOR gate. If one of its two inputs is high, but not both, it will produce a low output. An EXOR gate with a little circle on the output is the symbol. Inversion is represented by the little circle

XNOR gates/ Boolean Algebra

Laws of Boolean Algebra

In digital electronics, Boolean Algebra is a type of mathematical algebra that is utilized in digital logic. Algebra is a depiction of a statement in symbolic form (generally mathematical statements). In Boolean algebra, there are also expressions, equations, and functions.

Any logic design’s major goal is to make the logic as simple as possible so that the final implementation is simple. To simplify the logic, the Boolean equations and expressions that embody it must be simplified as well.

Some principles and theorems have been presented to simplify Boolean equations and expressions. It is quite straightforward to simplify or minimize the logical difficulties of any Boolean statement or function using these rules and theorems.

We analyze digital gates and circuits using Boolean Algebra. In an attempt to reduce the number of logic gates required, we can apply these “Laws of Boolean” to both reduce and simplify a complex Boolean expression.

As a result, Boolean Algebra is a logic-based mathematics system with its own set of rules and principles for defining and reducing Boolean expressions.

In Boolean Algebra, variables can only have one of two values: logic “0” or “1,” but an expression can contain an endless number of variables, each labeled uniquely to represent the expression’s inputs. Variables A, B, C, and so on are examples.

variables A, B, C, etc, giving us a logical expression of A + B = C, but each variable can ONLY be a 0 or a 1.

Examples of these individual laws of Boolean, rules, and theorems for Boolean Algebra are given in the following table.

In this article, Boolean algebra is used to show some of the most often utilized rules and theorems.

What is the use of Boolean Algebra?

Boolean algebra is the algebra subarea in which the values of the variables are true and false and normally denoted 1 or 0.
The principal algebra operators are or have not been used in set theory and statistics for Boolean algebra. Boolean algebra

Description of the Laws of Boolean Algebra

Law of Annulment – A term ANDed with a “0” equals 0 while a term ORed with a “1” equals 1.

A.0 = 0 A   determinant 0 is always equivalent to 0 when AND’ed with 0 1 Equals

A + 1= 1  A variable that is used with 1 always equals 1.

Identity Law – When a phrase is OR‘ed with a “0” or AND‘ed with a “1,” the result is always the same.

Identity Law – When a phrase is OR‘ed with a “0” or AND‘ed with a “1,” the result is always the same.

  • A + A = A    A variable OR’ed with itself is always equal to the variable.
  • A . A = A    A variable AND’ed with itself is always equal to the variable.

Complement Law – A term AND‘ed with its complement equals “0,” and a term OR‘ed with its complement equals “1,” 

  • A . A = 0    A variable AND’ed with its complement is always equal to 0.
  • A + A = 1    A variable OR’ed with its complement is always equal to 1.

Commutative law – Two separate terms are not important in the application order

A + B = A + B It makes no difference what order two variables are ANDed in.

A + B equals B + A The order in which two variables are OR’ed makes no difference.

The Double Negation Law – states that a term that has been inverted twice is equal to the original term.

A + A = A A variable’s double complement is always equal to the

De Morgan Theorem

Binary addition, binary subtraction, binary division, and binary multiplication are all part of Boolean algebra. Another key theorem on which the Boolean algebraic system is based is similar to these basic laws. That is the law of De Morgan.

De Morgan’s theorem is another name for this. This law operates on the basis of the Duality notion. Duality refers to the swapping of operators and variables in a function, such as replacing 0 with 1 and 1 with 0, and the OR operator with the AND operator.

There are two rules or theorems known as “de Morgan’s.”

(1) Two distinct terms NOR‘ed together is equivalent to the inverted terms (Complement) and AND‘ed together, for example, A + B = A.

(2) Two distinct terms NAND‘ed together is equivalent to the inverted terms (Complement) and OR‘ed together, for example, A.B. is equal to A + B.

The Duality Principle is extended by De Morgan’s law. De Morgan proposed two theorems to aid in the solution of algebraic issues in digital electronics.

The following are De Morgan’s statements:

Statement 1:

“The disjunction of the negations is the negation of the negations.” “The compliment of the product of two variables equals the sum of the compliments of separate variables,” we can say.

(A.B)’ = A’+B’

Statement 2:

“The negation of disjunction is the conjunction of the negations”. Or we can define that as “The compliment of the sum of two variables is equal to the product of the complement of each variable”.

(A + B)’ = A’.B’

Truth Tables The truth tables are a simple way to explain De Morgan’s laws.

The truth table for De Morgan’s first assertion ((A.B)’ = A’ + B’) is shown below.

Truth Table

A B A’ B’ A.B (A.B) A’+B’
0 0 1 1 0 1 1
0 1 1 0 0 1 1
1 0 0 1 0 1 1
1 1 0 0 1 0 0

As a result, De Morgan’s first law can be stated as “not (A and B) equals (not A) or (not B)”.

The truth table for De Morgan’s second assertion ((A + B) Equals A’.B’) is shown below.

A B A’ B’ A+B (A+B) A’.B’
0 0 1 1 0 1 1
0 1 1 0 1 0 0
1 0 0 1 1 0 0
1 1 0 0 1 0 0

So the De Morgan’s first law can also be expressed as “not (A or B) is equal to(not A) and (not B)”.

Conclusion

The mathematical system of boolean algebra is used to study logic circuits. To simplify Boolean expressions, boolean identities and algebraic manipulation can be used.

Simple circuits are the result of simplified boolean expressions.

  • AND, OR, and NOT gates are the most basic logic gates.
  • NAND and NOR gates are universal logic gates.
  • The XOR and XNOR gates are exclusive logic gates.

To know more about Boolean Algebra watch videos

Boolean Algebra
Boolean Algebra

Is Boolean algebra difficult?

Boolean Algebra is simple logic that becomes complicated once the problem is scaled up. In my case, I have learned Boolean Algebra for a Digital Circuits and Computer Hardware class. I personally found it is difficult once the tasks became more complex since our professor loved to get creative with his problems.

How do I find Boolean operators?

There is three Boolean commands search; AND, OR and NOT.
AND searches or finds all the search terms.
OR searches find one term or other. …
NOT eliminates items that contain the specified term.

Does Google use Boolean operators?

Boolean methods can be used in any search engine: Google, Linked In, or even on Facebook. Boolean is a term used to define the process of combining keywords with words called “operators.” These operators tell the search engine how to use the keywords in the search.

Print Friendly, PDF & Email
by Rajesh Bhatt
Rajesh is an IT Lecturer with a keen interest in Technological Research over the world. He is equally passionate about Gadgets, Music & Travelling.

2 thoughts on “Boolean algebra Simplification, examples, and solutions”

Leave a Comment

Hey, wait!Get Regular Update's

Subscribe Stay Connected, Keep Learning!