How to solve boolean algebra expressions? | AtomsTalk
Boolean algebra comes from algebra which is one of the main branches of mathematics. There are so many branches of mathematics like calculus, arithmetic, algebra, geometry, numerical analysis, etc., which are further developed in different categories. These branches are used in almost all fields and have different methods to solve their related problems.
To discuss about Boolean algebra we only go through its main branch viz. Algebra. There are five different types of algebra which are as follows:
- Abstract algebra
This type of algebra is used to solve the concepts regarding sets, binary operations, associativity, identity elements & inverse elements.
- Advanced Algebra
This branch helps you solve the problems related to equations with inequalities, conic sections, polynomial equations, sequences & series, matrices, quadratic functions with inequalities, rational expressions etc.
- Elementary algebra
It covers the concepts including evaluation of expressions and equations, variables, properties of inequalities and equalities etc.
- Linear algebra
The topics including vector spaces, relations and computations, linear equations and matrices & matrix resolution are covered under this branch.
- Communicative algebra
This branch studies algebraic number theory, algebraic geometry, etc.
Define Boolean algebra
Boolean algebra deals with the operations on logical values and its variable values are truth values only ie. true and false. Generally, it is denoted by 1 and 0. Boolean algebra is also called logical algebra or binary algebra and it is in the main branch of abstract algebra.
Operations performed on Boolean algebra
The following are the basic operations performed on Boolean algebra:
- NOT Operation
- OR Operations
- AND Operation
Not operations are also called negation and are represented as ¬ or !
The OR operation is known as Disjunction and is represented as ∨ or II
Where
AND operation is also defined as conjunction and is represented as ∧ or &&
Graphical representation of Boolean algebraic operations
Boolean algebra operations can be represented by Venn Diagrams by shading overlapping regions.
Venn diagram representing conjunction, disjunction and negation is as below:
Boolean expressions:
Boolean expressions are logical statements written using logical operators.
For example:
X OR Y
A AND B
(Here OR & AND are logical operators)
Truth table:
The truth table provides all possible logical values for logical variables and their combinations.
| A | B | A ∨ B | A ∧ B |
| True (1) | True (1) | True (1) | True (1) |
| True (1) | False (0) | True (1) | False (0) |
| False (0) | True (1) | True (1) | False (0) |
| False (0) | False (0) | False (0) | False (0) |
| A | ¬A |
| True (1) | False (0) |
| False (0) | True (1) |
The Laws of Boolean Algebra
There are six laws of Boolean algebra that are used to solve the Boolean algebraic expressions:
- AND teams
x . 0 = 0
x . 1 = x
x . x = x
x . xƟ = 0
- OR law
x + 0 = x
x + 1 = 1
x + x = x
x + xট = 1
- Inversion law
x + x̿ = 1
(This means that the result of double inversion of a variable is the original variable.)
- Associative law
(xy). z = x . (y . z)
(x + y) + z = x + (y + z)
- Distributive law
x . (y . z) = (x . y) + (x . z)
x+ (y . z) = (x + y). (x . z)
- Commutative law
x . y = y.x
x + y = y + x
Solving Boolean algebra expressions of De Morgan's law:
The most used theorems for solving Boolean algebra are De Morgan's laws. These are of two types: De Morgan's First Law and De Morgan's Second Law.
De Morgan's first team: (x . y)' = x' + y'
It states that the sum of their individual complements of a variable is equal to the complement of product of variables.
Truth table to verify the De morgan 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 |
Second Law by De Morgan: (x + y)' = x' . y'
It states that the complement of the sum of the variable is equal to the product of their individual complements or vice versa.
Truth table to verify the De Morgan 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 |
Boolean algebra expressions can be written in various forms and their results can be generated with truth tables by using a Boolean algebra calculator.
Example 1: Draw a truth table for x (y + z)
Solution:
Given expression is x (y + z)
| x | y | Z | y + z | x (y + z) |
| 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 1 | 1 | 1 |
| 1 | 1 | 0 | 1 | 1 |
| 1 | 1 | 1 | 1 | 1 |
| 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 |
| 0 | 1 | 0 | 1 | 0 |
| 0 | 1 | 1 | 1 | 1 |
Example 2: Draw a truth table for (x + y) (x + z)
Solution:
Given is (x + y) (x + z)
| x | y | z | x + y | x + z | (x + y) (x + z) |
| 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 1 | 0 |
| 0 | 1 | 0 | 1 | 0 | 0 |
| 0 | 1 | 1 | 1 | 1 | 1 |
| 1 | 0 | 0 | 1 | 1 | 1 |
| 1 | 0 | 1 | 1 | 1 | 1 |
| 1 | 1 | 0 | 1 | 1 | 1 |
| 1 | 1 | 1 | 1 | 1 | 1 |
You can verify this table with an online Boolean expression counter.
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