Mathematics has different branches like geometry, Statistics, Trigonometry, Arithmetic, and Algebra. Algebra has different sub-branches like elementary algebra, linear algebra, abstract algebra, and Boolean algebra.
Boolean algebra is the study of logic. In this article, we will study the definition, example, and real-life application of Boolean algebra.
What is Boolean algebra?
In Boolean algebra, we deal with the values of variables that have only two outputs 0 or 1.
0 is the value for a false statement and 1 is for the true one. Boolean algebra is also known as the algebra of logic.
Boolean algebra follows Aristotle’s logic. Non-Aristotle’s logic has more than two possibilities in output.
Proposition
The statement which we prove that it is true or false is called a proposition. The proposition has only two results. If the statement is true then our proposition is true and the numerical answer is 1. If the proposition is false then our result is 1.
The number of propositions tells us how many times a proposition true (1) and false (0) occur.
The Proposition is represented by Capital alphabets like P, Q, and S.
Let
If we have only one proposition. 21 = 2
If we have two propositions. 22 = 2*2= 4
If we have three propositions. 23 = 2*2*2 = 8
If we have n number of propositions. 2n= 2*2*2*…n times
Boolean algebra Operations
We will discuss some basic operations of Boolean algebra.
- AND operation / Conjunction
- Not operation / Negative
- OR operation / Disjunction
The symbol of these Operations are following
- Conjunction / AND of two propositions P and Q is the procedure the result true if both P and Q statements are true otherwise false.
- Disjunction / OR of two propositions P and Q is false only when both statements are false otherwise true if at least one proposition is true.
- Negative / NOT of a proposition makes the reverse the statement. It converts true into false and false into true.
What is the truth table?
The truth table is the representation of the propositions in the form of a table to analyzing our proposition is true or false according to the given expression.
Boolean algebra Truth Table
We express the above Boolean operators in the truth table;
P and Q are two propositions.
T represents the true.
F represents the false.
Laws of Boolean algebra
The list of laws of Boolean algebra is given as
- Inversion law
- OR law
- AND law
- Distribution law
- Commutative law
- Associative law
Try a Boolean algebra calculator to get the result according to the laws of binary algebra.
1. Inverse law
The inverse is defined as if we take the two times inverse of a proposition then the result is the original proposition.
It is denoted as (-(-P)) = P
2. OR law
When OR operation is used it follows the OR law.
- P+0 = P
- P+1 = 1
- P+P = P
- P+(-p) = 1
3. AND law
When AND operation is used it follows the AND law.
4. Distribution law
The distribution law state
5. Commutative law
Commutative law state as the changing of the position of the variable does not affect the result.
6. Associative law
If we have three propositions P, Q and R are defined as
Conditional Operation
When two propositions follow the if-then operator. It gives the only false answer when the 2nd proposition is false. Otherwise always true.it is denoted by .
Now we consider more two Boolean algebra on the bases of results.
- Tautology
- Contradiction
1. Tautology
In tautology, all the results just give us the true value.
2. Contradiction
If the result of the truth yields all the output values wrong then it is called a contradiction.
Example:
Find Boolean expression (-P*(-P+Q)) +Q) by using the truth table.
Solution:
Step 1: Take two propositions P and Q.
Step 2: Find NOT of P.
Step 3: Find the NOT of P OR Q. symbolically it is represented by (-P+Q).
Step 4: Find AND of P with (-P+Q).
Step 5: Find the NOT of (P*(-P+Q)).
Step 6: Find AND of Q with – (P*(-P+Q)).
Application of Boolean algebra
- Light Switch
- Probability
- geometry of sets
- In computers as Information theory
Summary
In this article, we have learned about the definition, examples, truth table, and applications of Boolean algebra. Now after reading the above post, you can grab all the basics of Boolean algebra.
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