Boolean Algebra
Boolean algebra is a branch of algebra dealing with logical operations on variables. There can be only two possible values of variables in boolean algebra, i.e. either 1 or 0. In other words, the variables can only denote two options, true or false. The three main logical operations of boolean algebra are conjunction, disjunction, and negation.
In elementary algebra, mathematical expressions are used to mainly denote numbers whereas, in boolean algebra, expressions represent truth values. The truth values use binary variables or bits "1" and "0" to represent the status of the input as well as the output. The logical operators AND, OR, and NOT form the three basic boolean operators. In this article, we will learn more about the definition, laws, operations, and theorems of boolean algebra.
What is Boolean Algebra?
Boolean algebra is also known as binary algebra or logical algebra. The most basic application of boolean algebra is that it is used to simplify and analyze various digital logic circuits. Venn diagrams can also be used to get a visual representation of any boolean algebra operation.
Boolean Algebra Definition
Boolean algebra can be defined as a type of algebra that performs logical operations on binary variables. These variables give the truth values that can be represented either by 0 or 1. The basic Boolean operations are conjunction, disjunction, and negation. The logical operators AND, OR, and NOT are used to represent these operations respectively. Furthermore, these operations are analogous to intersection, union, and complement of sets in set theory. Some of the Boolean algebra rules are:
- Any variable that is being used can have only two values. Binary 1 for HIGH and Binary 0 for LOW.
- Every complement variable is represented by an overbar i.e. the complement of variable B is represented as B¯. Thus if B = 0 then B¯= 1 and B = 1 then B¯= 0.
- Variables with OR are represented by a plus (+) sign between them. For example OR of A, B, C is represented as A + B + C.
- Two or more variables with logical AND are represented by writing a dot between them such as A.B.C. Sometimes the dot may be omitted like ABC.
Let us look at an example,
Suppose we have two variables A = 1 and B = 0. We have to perform the AND operation. The boolean expression can be represented as A.B = 1.0 = 0.
If we have to perform the logical OR operation then the boolean expression is given as A + B = 1 + 0 = 1.
If we apply the NOT operation on both the input variables then we get \(\overline{A}\) = 0 and \(\overline{B}\) = 1.
Boolean Algebra Expression
Boolean algebra expressions are statements that make use of logical operators such as AND, OR, NOT, XOR, etc. These logical statements can only have two outputs, either true or false. In digital circuits and logic gates "1" and "0" are used to denote the input and output conditions. For example, if we write A OR B it becomes a boolean expression. There are many laws and theorems that can be used to simplify boolean algebra expressions so as to optimize calculations as well as improve the working of digital circuits.
Boolean Algebra Operations
There are three basic Boolean algebra operations. These are conjunction, disjunction, and negation. The equivalent logical operators to these operations are given below.
- AND operator: It is analogous to conjunction. In a boolean expression, "•" is used to represent the AND operator. This operator returns true if and only if all input operands are true.
- OR operator: This operator is equivalent to disjunction. In a boolean expression, "+" symbol is used to represent the OR operator. The operator returns true if and only if one or more of the input operands are true.
- NOT operator: This logical operator is comparable to negation. NOT operator returns true if the input variable is false. Similarly, if the input variable is false it returns true. An overline on the variable is used to represent this operator.
Boolean Algebra Laws
The main use of boolean algebra is in simplifying logic circuits. By applying Boolean algebra laws, we can simplify a logical expression and reduce the number of logic gates that need to be used in a digital circuit. Some of the important boolean algebra laws are given below:
Distributive Law
The distributive law says that if we perform the AND operation on two variables and OR the result with another variable then this will be equal to the AND of the OR of the third variable with each of the first two variables. The boolean expression is given as
A + B.C = (A + B) (A + C)
Thus, OR distributes over AND
If we OR two variables then AND their result with another variable then this value will be equal to the OR of the AND of the third variable with the other two variables. This is given by:
A .(B+C) = (A.B) + (A.C)
Hence, AND distributes over OR.
Associative Law
According to the associative law, if more than two variables are OR'd or AND'd then the order of grouping the variables does not matter. The result will always be the same. The expressions are given as:
A + (B + C) = (A + B) + C
A.(B.C) = (A.B).C
Commutative Law
Commutative law states that if we interchange the order of operands (AND or OR) the result of the boolean equation will not change. This can be represented as follows:
A + B = B + A
A.B = B.A
Absorption Law
Absorption law links binary variables and helps to reduce complicated expressions by absorbing the like variables. There are 4 statements that fall under this law given as:
- A + A.B = A
- A (A + B) = A
- A + Ā.B = A + B
- A.(Ā + B) = A.B
There are some boolean algebra properties and identities that are given as follows:
- A + 1 = 1
- A + 0 = A
- A . 1 = A
- A . 0 = 0
- A + A = A
- A . A = A
- \(\overline{\overline{A}}\) = A
- A + \(\overline{A}\) = 1
- A . \(\overline{A}\) = 0
Boolean Algebra Theorems
One of the most important theorems in boolean algebra is de morgan's theorem. This theorem comprises two statements that help to relate the AND, OR, and NOT operators. The two statements are given as follows:
- When two variables are AND'd and negated the result is equal to the OR of each negated input variable. The boolean expression is \(\overline{A.B}\) = \(\overline{A}\) + \(\overline{B}\).
- When two variables are OR'd and negated then this will be equal to the AND of each negated input variable. This is given by \(\overline{A + B}\) = \(\overline{A}\).\(\overline{B}\)
Boolean Algebra Postulates
Boolean algebra postulates are not laws or theorems but are statements that hold true. These postulates are the four possible logical OR and logical AND operations as well as the rules followed by the NOT operator. Given below are the boolean algebra postulates:
- 0 + 0 = 0
- 0 + 1 = 1
- 1 + 0 = 1
- 1 + 1 = 1
- 0 . 0 = 0
- 0 . 1 = 0
- 1 . 0 = 0
- 1 . 1 = 1
- \(\overline{1}\) = 0
- \(\overline{0}\) = 1
Boolean Algebra and Logic Gates
A logic gate is a building block for any digital circuit. These logic gates need to make the decision of combining various inputs according to some logical operation and produce an output. Logic gates perform logical operations based on boolean algebra. Suppose we have two inputs A and B. Let the output be R. Then given below are the various types and symbols of logic gates.
AND gate - R = A.B will be the boolean expression. R will be true if both A AND B are true.
OR gate - The boolean equation is R = A + B. Here, R will be true if either of the inputs A OR B is true.
NOT gate - This is also known as an inverter and the boolean equation is R = \(\overline{A}\). This implies that the output is true only if the input is false.
NAND gate - This is also the NOT - AND gate. R = \(\overline{A.B}\) is the boolean equation. The output R will NOT be true if both A AND B are true.
NOR gate - The NOT - OR operation results in the NOR gate. R = \(\overline{A + B}\) denotes the boolean equation and implies that R is true if A and B are NOT true.
EX - OR gate - This is the exclusive OR gate. It can be created by using a combination of the above-mentioned gates. R = A ⊕ B is the boolean expression. It means that R is true only if either A or B is true.
EX - NOR gate - The boolean equation of the exclusive NOR gate is given as R = \(\overline{A ⊕ B}\). It means R is true only if both inputs are either true or false.
Boolean Algebra Truth Table
Boolean algebra truth table can be defined as a table that tells us whether the boolean expression holds true for the designated input variables. Such a truth table will consist of only binary inputs and outputs. Given below are the truth tables for the different logic gates.
AND gate
| A | B | R = A.B |
| 0 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
OR gate
| A | B | R = A + B |
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 1 |
NOT gate
| A | R = \(\overline{A}\) |
| 1 | 0 |
| 0 | 1 |
NAND gate
| A | B | R = \(\overline{A.B}\) |
| 0 | 0 | 1 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
NOR gate
| A | B | R = \(\overline{A + B}\) |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 0 |
EX - OR gate
| A | B | R = A ⊕ B |
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
EX - NOR gate
| A | B | R = \(\overline{A ⊕ B}\) |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
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Boolean Algebra Examples
-
Example 2: Simplify the logic circuit using boolean algebra.
Solution: Output of gate.
On solving the expression AB + BC(B + C) = AB + BBC + BCC.
Applying A.A = A we get AB + BC + BC.
Applying A + A = A we get AB + BC = B(A + C).
Answer: B(A + C)
FAQs on Boolean Algebra
What is Meant by Boolean Algebra?
Boolean algebra is a type of algebra where the input and output values can only be true (1) or false (0). Boolean algebra uses logical operators and is used to build digital circuits.
What are the Boolean Algebra Rules?
The basic boolean algebra rules are:
- Any variable that is being used can have only two values. Binary 1 for HIGH and Binary 0 for LOW.
- Every complement variable is represented by an overbar i.e. the complement of variable B is represented as B¯. Thus if B = 0 then B¯= 1 and B = 1 then B¯= 0.
- Variables with OR are represented by a plus (+) sign between them. For example OR of A, B, C is represented as A + B + C.
- Two or more variables with logical AND are represented by writing a dot between them such as A.B.C. Sometimes the dot may be omitted like ABC.
What are the Boolean Algebra Laws?
There are four main laws of boolean algebra. These are distributive law, associative law, commutative law, and absorptive law. When we simplify boolean expression these laws are extensively used.
What are the Identities of Boolean Algebra?
The important boolean algebra identities are given below:
- A + 1 = 1
- A + 0 = A
- A . 1 = A
- A . 0 = 0
- A + A = A
- A . A = A
- \(\overline{\overline{A}}\) = A
- A + \(\overline{A}\) = 1
- A . \(\overline{A}\) = 0
How Do you Do Boolean Algebra?
When solving a boolean algebra expression the most important thing is to remember the boolean algebra laws, theorems, and associated identities. We need to consecutively apply these rules until the expression cannot be simplified further to get our answer.
What is the Distributive Law in Boolean Algebra?
There are two statements under the distributive law in boolean algebra. The two statements are as follows:
- OR distributes over AND [A + B.C = (A + B) (A + C)].
- AND distributes over OR [A .(B+C) = (A.B) + (A.C)]
What is Absorption Law in Boolean Algebra?
The boolean algebra equations for the absorption law that help to link like variables are as follows:
- A + A.B = A
- A (A + B) = A
- A + Ā.B = A + B
- A.(Ā + B) = A.B
How to Simplify Boolean Algebra Expressions?
We can simplify boolean algebra expressions by using the various theorems, laws, postulates, and properties. In the case of digital circuits, we can perform a step-by-step analysis of the output of each gate and then apply boolean algebra rules to get the most simplified expression.
What are the Properties of Boolean Algebra?
Boolean algebra has three basic properties, they are:
- Commutative Property of Addition and Multiplication: Order of variables can be reversed without changing the truth of expression i.e. A + B = B + A and AB = BA
- Associative Property of Addition and Multiplication: Multiplied and added variables together with parentheses can be altered without changing the truth of expression i.e. A + (B + C) = (A + B) + C and A(BC) = (AB)C
- Distributive Property: Expression formed by the product of a sum when expanded and reversed shows how the terms may be factored i.e. A(B + C) = AB + BC