Truth Table

What is the 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 on the basis of 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 their respective columns  that list all the possible true 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.


X

Y

AND

(∧)

OR

(∨)

NOR

(~∨)

NAND

(~∧)

XOR

(⊻)

Conditional

(⇒)

Biconditional

(⇔)

T

T

T

T

F

F

F

T

T

T

F

F

T

F

T

T

F

F

F

T

F

T

F

T

T

T

F

F

F

F

F

T

T

F

T

Y


In the above table T indicates true and F indicates False


Let us now discuss each binary operations 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 statement which 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 are 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

x

y

x→y

y→x

(x→y)^(y→x)

T

T

T

T

T

T

F

F

T

F

F

T

T

F

F

F

F

T

T

T


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

x

y

x→y

T

T

T

T

F

F

F

T

F

F

F

T


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

y ∧ z∧ ¬x

T

T

T

F

T

T

F

F

T

F

T

F

T

F

F

F

F

T

T

T

F

T

F

F

F

F

T

F

F

F

F

F


  1. Examine the following contingent statement.

x ∨ ¬ y ∨ ¬ z


What would be the truth table for the above statement?


x

y

z

x ∨ ¬ y ∨ ¬ z

T

T

T

T

T

T

F

T

T

F

T

T

T

F

F

T

F

T

T

F

F

T

F

T

F

F

T

T

F

F

F

T


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


X

Y

Output

T

T

F

T

F

T

F

T

T

F

F

F

FAQ (Frequently Asked Questions)

1. What is Known as Boolean Algebra?

Boolean Algebra is the classification of algebra in which the values of the variables are the true values, true and false usually represented as 0 and 1 respectively. It is used to examine and simplify digital circuits. It is also known as binary algebra or logical algebra. It is fundamentally used in the development of digital electronics and is provided in all the modern programming languages. 


The important operations carried out in boolean algebra are conjunction (∧), disjunction (∨), and negation (¬). 


Hence, the boolean algebra is quite different from elementary algebra where the values of the variables are numerical and arithmetic operations such as addition, subtraction is also executed on them.