Truth Table (original) (raw)
Last Updated : 23 Jul, 2025
A Truth Table is a table that lists all the possible combinations of inputs and their corresponding outputs. It shows how the output of logic circuits changes with different combinations of logic levels at the input. It is mostly associated with Boolean algebra or areas where Boolean logic is used. It is a branch of algebra where there are only two values possible true and false.
Table of Content
- Truth Table
- Construction
- Unary Operation Truth Table
- Binary Operations Truth Table
- Applications
- Advantages
- Disadvantages
What is Truth Table?
The truth table is a systematic representation of all truth values of a logical expression. It consists of all the inputs and gives the required output. It takes a combination of inputs to get the desired output. Truth tables are mainly used in Boolean algebra so, a variable can take two values 0 or 1.
The truth table is primarily used in digital circuits where it is used to validate the output generated from the various input combinations of the logical expressions. It is also used in mathematics and other fields which use Boolean logic and digital logic.
How to Create a Truth Table for Given Logical Expression
A truth table has some elements that include Input Variables, Output Variables, and combinations of input variables. The columns represent the number of input and output variables, and the rows represent the number of all possible combinations. As in the truth table, we generally work with binary there will be 2n number of rows, where n is the number of input variables.
Steps involving the construction of the truth table are:
- Identification of input variables and determining the number of rows.
- Listing all the possible combinations.
- Determining the output for each combination using the logical expression.
Truth Table for Unary Operation
A Unary operation is a kind of operation where there is only one operand present. There are four types of unary operation:
- Logical True
- Logical False
- Logical Identity
- Logical Negation
Logical True
In this operation, whatever the input is the output will be always true. The truth table is as follows:
| INPUT | OUTPUT |
|---|---|
| True | True |
| False | True |
Logical False
In this operation, whatever the input is the output will be always False. The truth table is as follows:
| INPUT | OUTPUT |
|---|---|
| True | False |
| False | False |
Logical Identity
In this operation, whatever the input is the output will remain the same. This is also known as the NULL operator. The truth table is as follows:
| INPUT | OUTPUT |
|---|---|
| True | True |
| False | False |
Logical Negation
In this operation, whatever the input is the output will be opposite of the input. For true input, it will return False and vice-versa. This is also known as the NOT operation. The truth table is as follows:
| INPUT | OUTPUT |
|---|---|
| True | False |
| False | True |
Truth Table for Binary Operations
A binary operation is a kind of operation where two operands are involved. Using two operands and a single operator which gives a single output. Some of the basic binary operations include AND, OR, NAND, NOR, XOR, conditional and biconditional.
AND & NAND Operation
AND gate is one of the basic operations of Boolean algebra that gives an output is 1 when all inputs are 1 otherwise the output is 0. The operation between two inputs will be A.B which is read as A AND B.
The NAND (NOT AND) is also a binary operator that is used as a universal logic gate and can be used to implement other gates. If at least one of the inputs is 0 the output is 1 else the output is 0. The operation between two operands will be A ⊼ B which is read as A NAND B.
| A | B | C = A ∧ B | C= A ⊼ B |
|---|---|---|---|
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 1 |
| 1 | 0 | 0 | 1 |
| 1 | 1 | 1 | 0 |
OR and NOR Operation
OR gate is also a basic operation of Boolean algebra that gives an output is 1 when at least one input is 1 otherwise the output is 0.o 0. The operation between two inputs will be A + B or A ∨ B which is read as A OR B.
The NOR (NOT OR) is also a binary operator that gives output 1 when both the inputs are 0 else output is 1. The operation between two operands will be A ⊽ B which is read as A NOR B.
| A | B | C = A ∨ B | C = A ⊽ B |
|---|---|---|---|
| 0 | 0 | 0 | 1 |
| 0 | 1 | 1 | 0 |
| 1 | 0 | 1 | 0 |
| 1 | 1 | 1 | 0 |
XOR Operator
The XOR (Exclusive or) is another binary operator that gives output 1 when all the inputs are different else the output is 0. The operation between two operands will be A⊕B which is read as A XOR B.
| A | B | C = A ⊕ B |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
XNOR Operator
The exclusive NOR operator i.e., XNOR gives the output as 1 when both the operands are same and output as 0 when both the operands are different. It is the NOT operation of XOR operation.
| A | B | C = A XNOR B |
|---|---|---|
| 0 | 0 | 1 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
Conditional and Bi-conditional Operation
A conditional operation is one of the logical operations that is used to represent conditional statements or "if-then" statements or implications. The output of the operation is 0 only when the first input is 1 and the second is 0 and in all other cases the output is 1. Its symbol is A → B.
A biconditional operation is a binary operation that represents "if and only if". If both inputs are either 1 or 0 then the output will be 1 and if both inputs are different then the output is 0. It is denoted by A↔ B.
| A | B | C = A → B | C = A ↔ B |
|---|---|---|---|
| 0 | 0 | 1 | 1 |
| 0 | 1 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 1 | 1 | 1 | 1 |
Solved Examples of Truth Table
**Example 1: Draw the truth table of the given Boolean expression: A.(B+C)
**Solution:
The number of input combinations will be 8 (23=8).
| A | B | C | B+C | A.(B+C) |
|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 |
| 0 | 1 | 0 | 1 | 0 |
| 0 | 1 | 1 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 1 | 1 | 1 |
| 1 | 1 | 0 | 1 | 1 |
| 1 | 1 | 1 | 1 | 1 |
**Example 2: Draw the truth table of the given Boolean expression: A.¬ (B + (C.D))
**Solution:
There are four inputs so the number of input combinations will be 24=16
| A | B | C | D | C.D | B+(C.D) | ¬(B+(C.D)) | A.¬(B+(C.D)) |
|---|---|---|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 1 | 0 | 1 | 0 | 0 |
| 0 | 1 | 1 | 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 1 | 1 | 1 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 |
| 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 |
| 1 | 0 | 1 | 0 | 0 | 0 | 1 | 1 |
| 1 | 0 | 1 | 1 | 1 | 1 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 1 | 0 | 0 |
| 1 | 1 | 0 | 1 | 0 | 1 | 0 | 0 |
| 1 | 1 | 1 | 0 | 0 | 1 | 0 | 0 |
| 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 |
Applications of Truth Table
Some applications of truth tables are listed below.
- The most important use of truth table is in logic design of digital circuits. It is used for the design and validation of various logical expressions and then for optimization of digital circuits.
- Truth table is also used in computer programming, it is used to design and analyze the logical behavior of algorithms. It plays an important role in the use of bitwise operators.
- In communication systems, it is used for error detection. Logical conditions are analyzed to identify and correct the codes with the help of a truth table.
- In mathematics, truth tables are used in set theory for an understanding of the relationship between sets using logical operations and it is also used to simplify logical expressions in predicate logic.
Advantages of Truth Table
Some advantages of the truth table are listed below.
- The truth table provides clean representation which makes it easier to analyze.
- It provides a systematic method for analyzing all possible combinations of truth values of a particular logical expression.
- It is very easy to implement.
- It helps to find the various outcomes of different logical conditions and hence helps in design.
Disadvantages of Truth Table
Some disadvantages of the truth table are listed below.
- As the number of variables increases it also grows in size and becomes very big making it difficult to use.
- It provides redundant information sometimes when the logical expression follows a pattern.
- It is very time-consuming for big logical expressions as the number of variables is more.
- It cannot be used for optimization.
- It is highly dependent on expressions, and it is difficult to convert language statements into truth tables.
Conclusion
In conclusion, a truth table is a table that is used to find the output of logical expressions for its various combinations of input. Truth table is widely used in digital circuits along with this it is also used in other fields like computer programming and mathematics.
It is used in various fields because of its easy-to-implement method along with systematic and clear representation making it easy to interpret. Overall, the truth table is one of the most important components for design of digital circuits and in Boolean algebra.