Boolean algebra symbols are essential to be studied if we want to use and understand the boolean algebra and logic states. Boolean algebra is mainly used when dealing with logic gates such as JK flip-flop.
What is Boolean Algebra
Boolean algebra is one of the mathematical branches that excels at logic states, equations, variables, and operations.
Since it is used for logic operations, we will only see two conditions:
- True (1)
- False (0)
These two values can be used freely to form a complex equation for logical processing.
We will manipulate the (1) and (0) along with common mathematical operators, addition (+) symbol used for OR, dot product or multiplication (•) symbol used for AND, and negation (NOT or ‘) symbol to invert the value. Alphabetical letters (A, B, C, etc) represent our variables and their values are only 0 or 1.
Boolean Operator Precedence
Just like basic mathematical precedence where we prioritize multiplication and division over addition and subtraction, Boolean algebra also has its own operator precedence. The Boolean operators symbols and their precedence are
- NOT (! or ‘)
- AND (•)
- OR (+)
For example if we have an expression A + B • C’ then
- First we have to invert the value of C.
- Second we have to do AND between B and C.
- Last we do OR the value of A with the result of the second step.
Not only boolean operator precedence, we can make our precedence with a pair of parentheses. With this, we can override the operator precedence. This is very helpful if we want to use operators but freely modify its processing steps.
Using the example above, if we add a pair of parentheses to A and B,
The addition of A and B will be executed first, followed by negation to the value of C, and lastly multiply the addition result with the negation result.
Boolean Algebra Symbols
Boolean or binary or logical algebra only has values of True (1) and False (0). Just as basic logical operation, boolean algebra also has boolean logic symbols:
- Conjunction (∧ or •)
- Disjunction (∨ or +)
- Negation (⎺ or ! or ‘)
Conjunction
The conjunction will act as an AND logic where it only produces True (1) if all the elements are True (1). The illustration of conjunction can be seen below.
We can express the illustration above with A ∧ B where the “True” condition is the area where A and B intersects each other.
(A ∧ B) only results True if both are True (1), or else the result will be False.
The basic mathematic of (A ∧ B) or (A • B) is
Disjunction
The disjunction will act as an OR logic where it only produces True (1) if one of the elements is True (1). The illustration of disjunction can be seen below.
We can express the illustration above with A ∨ B where the “True” condition is the all the area where A and B exist.
(A ∨ B) results True if one of them True (1), or else the result will be False.
The basic mathematic of (A ∨ B) or (A + B) is
Negation
This one doesn’t have to be explained long, since we only invert the value of a variable. The illustration of negation can be drawn below.
For example if we have (A = 1), then (A’ = 0).
Boolean Algebra Truth Tables
We can unify the three operations above in a single table as can be seen below. Here we will use “1” to represent True and “0” to represent False.
Boolean Algebra Laws
There are six Boolean algebra laws we must follow and they are not different from the basic mathematical calculation. You can see what we mean below.
- Commutative
- Associative
- Distributive
- AND
- OR
- Inversion
Commutative
Commutative law states that
Changing the sequence of the variables does not change its result.
Associative
Associative law states that
The order of the logic operations are irrelevant if they are the same.
Distributive
Distributive law states that
Multiplying a number by a group of numbers added together is not different from multiplying each of them separately.
AND
This law uses an AND operation or a multiplication operation.
OR
This law uses an OR operation or an addition operation.
Inversion
Inversion is to invert the value of a variable. Double inversion will make a variable have its own original value.
Boolean Algebra Theorems
Just like electrical circuit analysis where we have a lot of theorems to help us, Boolean algebra also has two strong theorems to simplify our job. The theorems are:
- De Morgan’s First Law
- De Morgan’s Second Law
These De Morgan’s laws are able to reduce the given Boolean expression into a simplified one.
De Morgan’s First Law
De Morgan’s First Law states that
The complement of the product of a group of variables is equal to the sum of their respective complement variables.
We can prove De Morgan’s First Law with the table below.
The last two columns are the conclusion of De Morgan’s First Law.
De Morgan’s Second Law
De Morgan’s Second Law states that
The complement of the sum of a group of variables is equal to the product of their respective complement variables.
We can prove De Morgan’s Second Law with the table below.
The last two columns are the conclusion of De Morgan’s Second Law.
