Truth Table

A truth table is a mathematical table used to carry out logical operations in Maths. It includes boolean algebra or boolean functions. It is primarily used to determine whether a compound statement is true or false based on the input values. Each statement of a truth table is represented by p,q or r and also each statement in the truth table has its respective columns that list all the possible truth values. The output which we get is the result of the unary or binary operations executed on the input values. Some of the examples of binary operations are AND, OR, NOR, XOR, XNOR, etc.

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Truth Table for Binary Operations

The binary operations include two variables for input values. Here, the output result relies on the operation executed on the input or proposition values and the value can be either true or false. Some of the major binary operations are:

• And

• Or

• NAND

• NOR

• XOR

• Biconditional 

• Conditional or “if-then” 

Now, we will construct the consolidated truth table for each binary operation, taking the input values as X and Y.

In the above table T indicates true and F indicates False

Let us now discuss each binary operation mentioned above

NOR and OR Truth Table Operation

OR statements represent that if any two input values are true. The output result will always be true. It is represented by the symbol ().

Whereas the NOR operation delivers the output values, opposite to OR operation. It implies that a statement that is true for OR, is false for NOR and it is represented as (~∨).

NAND and AND Operational True Table

From the above and operational true table, you can see, the output is true only if both input values are true, otherwise, the output will be false. In the and operational true table, AND operator is represented by the symbol (∧).

XOR Operation Truth Table

The table defines, the input values should be exactly either true or exactly false.  The symbol for XOR is represented by (⊻).

Conditional and Biconditional Truth Tables

Let x and y be two statements and if “ x then y” is a compound statement, represented by x → y and referred to as a conditional statement of implications. This implication x→y is false only when x is true and y is false otherwise it is always true. In this implication, x is known as antecedent or hypothesis and y is known as the conclusion or consequent.

Conditional Truth Table

In the above conditional truth table, when x and y have similar values, the compound statement (x→y) ^ (y→x) will also be true. If we combine two conditional statements, we will get a biconditional statement.

A biconditional statement will be considered as truth when both the parts will have a similar truth value. The conditional operator is represented by a double-headed arrow ↔.  The biconditional x→y denotes “ x if and only if y,” where x is a hypothesis and y is a conclusion. The table given below is a biconditional truth table for x→y.

Biconditional Truth Table

In the above biconditional truth table, x→y is true when x and y have similar true values ( i.e. either both x and y values are true or false).

Solved Examples

1. Examine the following contingent statement.

y ∧ z∧ ¬x

What would be the truth table for the above statement?

x ∨ ¬ y ∨ ¬ z

What would be the truth table for the above statement?

Quiz Time

1. The symbol ‘∧’ represent

1. and

2. or

3. not

4. Implies
   
   

2. The symbol ‘ ∨ ’ represent

1. and

2. or

3. not

4. Implies
   
   

3. Which type of logic is below the table show?

1. And

2. Or

3. Not

4. XOR
   
   

Importance of Learning the Truth Table for Binary Operations  

Learning the Truth Table – For Binary Operations is important as it is an essential part of mathematics. Binary operations require a lot of time and practice. However, with the use of the truth table, you can simplify these operations and solve the questions with ease. The main aim of making a truth table is to identify whether two or more statements are equivalent to each other or not. This concept is a little trickier to understand if you only refer to its definition. You have to go through solved examples and practice questions to understand the truth table perfectly. Below are some reasons why you should not skip the Truth Table – For Binary Operations.

• It is a scoring topic and holds a major portion of the marking distribution in the exam. So, by learning it, you can score well in your exam. 

• A truth table is not only used for binary operations. It is used to prove logical equivalencies and in digital logic, circuitry to determine the function of look-up tables. So, learning this concept will give you a better understanding of these topics as well.  

• Once you have learned the concept of Truth Table – For Binary Operations, it will become much easier for you to solve questions based on binary operations.  

How to Learn the Truth Table for Binary Operations

You have to spare an ample amount of time to learn the concept of the Truth Table – For Binary Operations. With the study materials provided by Vedantu, you will have a strong grasp of the Truth Table in no time. We provide you with a wide variety of resources to help you understand the concept more clearly.

Below are some tips that will aid you in learning this crucial concept: 

• Once you have read the definitions of the Truth Table, go through the solved examples and questions to get a better understanding of these tables.

• Use reference books to gain more knowledge. These books contain detailed explanations and solved examples to make it easier for you to learn the concept. 

• You can use sample papers and previous year questions papers to explore more questions based on the Truth Table for practice.