Introduction:
Boolean Algebra is a mathematical language of digital systems.
To design digital logic systems, the study of Boolean Algebra is imperative. It is one of the primitive tools for a logic designer, to design, analyze and simplify the logic.
Outline:
- To learn some Boolean algebra rules, laws and theorems.
- How to find the complement of a function
- Examples and explanation
- Applications of Boolean Algebra in digital electronics Or why do we need to learn Boolean Algebra
Three Basic Gates | Three Basic Operations In Boolean Algebra:
Boolean Inversion | The NOT Operator:
It performs an inversion operation. The output is the inverse of the input. Get to know about NOT operators, please check out my tutorial here.
Boolean Multiplication | The AND Operator:
It performs a Boolean multiplication operation. The laws and rules for multiplication are similar to elementary algebra. Get to know about NOT operators, please check out my tutorial here.
Boolean Addition | The OR Operator:
Boolean addition is equivalent to the OR operation
Laws Of Boolean Algebra:
The fundamental laws of arithmetic are also valid here. These laws are here
- Associative law
- Commutative law
- Distributive law
Commutative Law:
Commutative law for addition:
Commutative law for two variables is given below:
A + B = B + A
Commutative law for multiplication:
Commutative law for two variables is given below:
A . B = B . A
Associate Law:
Associative law for addition:
The associative law for three variables is given below:
A + (B + C) = (A + C) + B
Associative law for multiplication:
The associative law for three variables is given below:
A . (B . C) = (A . C) . B
Distributive Law:
The distributive law for three variables is given below:
(A + B) . C = A.C + B.C
Boolean Algebra Rules:
For practical digital systems, many complex logic blocks appear during the analysis. There are some rules of solving and simplifying logic expressions, which are almost the same as that of arithmetic rules. The topic is rather easy as the rules and laws are almost the same. Let’s have a look at them.
A + 0 = A
This is the additive identity.
For a two-input OR gate, one input is zero and the variable A has only two values (either 1 or 0). The output is equal to A.
To prove this rule, let’s have a look at the truth table.
| A | 2nd Input | Output |
| 0 | 0 | 0 |
| 1 | 0 | 1 |
2. A + 1 = 1:
= (A + 1) . 1 Rule 4
= (A + 1) . (A + AC). Rule 6
= (A + 1 . AC) Rule 12
= (A + AC) Rule 4
= 1 Rule 6
A . 0 = 0
This is a duality property.
A . 1 = A
This is multiplicative identity. Have a look at Rule 1, we will prove this rule in the same manner.
A + A = A
Proof:
A + A = (A + A) . 1 Rule 4
= (A + A) . (A + AC ) Rule 6
= A + A . AC Rule 12
= A + 0 Rule 8
= A Rule 4
A + AC = 1
This rule is verified with the help of a truth table. The right-hand side is equal to the left-hand side for both values of A.
| A | AC | Output = 1 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
A . A = A
Proof:
A . A = A . A + 0 Rule 1
= A . A + A . AC Rule 8
= A . (A + AC) Distributive Law
= A . 1 Rule 6
= A
A . AC = 0
This rule is verified with the help of a truth table.
| A | AC | Output = 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
(AC)C = A
Double negation or double inversion. It is easily seen in schematic diagram.
- A + A . B = A
= A . 1 + A . B Rule 4
= A . (1 + B) Distributive Law
= A . 1 Rule 2
= A Rule 4
A + ACB = A + B
= (A + A . B) + AC.BRule 10
= A + B . (A + AC) Distributive Law
= A + B . (1) Rule 6
= A + B Rule 4
(A + B)(A+C) = A + BC
= A . A + A . C + A . B + B .C Distributive law
= A + A . C + A . B + B .C Rule 7
= A . (1 + C) + A . B + B .C
Distributive law
= A . 1 + A . B + B .C Rule 2
= A . (1 + B) + B . C Distributive law
= A .1 + B . C Rule 2
= A + B . C Rule 4
Here some of more Boolean Expression exercises.
DeMorgan’s Theorem:
The two statements of DeMorgan’s Theorem is written below:
” The complement of a product of variables is equal to the sum of the complements of the variables.”
\( \overline{AB} = \overline {A}+ \overline B \)
“The complement of a sum of two variables is equal to the product of the complements of the variables.”
\(\overline {A + B} = \overline {A}. \overline {B}\)We have learnt DeMorgan’s theorem earlier. In the context of digital electronics, we will see it as gate and truth table form.
Frequently Asked Questions:
What is Boolean algebra with an example?
What is Boolean and example?
EWhat jobs use Boolean algebra?
Expand question
Is Boolean algebra easy?
What is a real life example of Boolean algebra?
Which is better, SOP or POS?
Why in Boolean algebra is 1 1 1?
Why does 1×1 equal 1?
What is 1 plus 1 in Boolean algebra?
What is the logic behind 1 1 11?
What does a ∧ b mean?
Why do we write 1 as 1?
Is Boolean algebra hard to learn?
What’s the hardest thing in algebra?
Does AI use Boolean algebra?
What is the easiest algebra class?
Is Boolean algebra calculus?
What is an example of a Boolean in Python?
What is == in Boolean?
What is Boolean with example?
Why do we use Boolean in Python?