Truth Tables and Logical Statements

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Introduction to Truth Table

Truth Table is a mathematical operation which is related to Boolean Algebra and calculus. It can also be termed as an interdisciplinary tool. The truth table is not only related to mathematics but also with philosophy and computer science. The actual concept is the same in every subject; however, the inscription or notation varies depending on the subject you are dealing with.


What are The Rules of Truth Table?

The statements in a truth table are represented by the letters or variables like p, q and r. These variables have columns and it states the possible value of truths. Let's show you how we will create a truth table by following the truth tables rules. Let us understand this by truth table examples.

Let us consider a sentence or statement like, "Disha is absent today". We are representing this sentence with the variable "p". It is either incorrect or correct, which means true or false.

p = Disha is absent today.

p can be true if Disha is absent, but if she isn't absent, then Disha is not absent.

The negation or negative word in the sentence is known as not p. Not p opposes with p and possesses the opposite truth value.

Not p = Disha is not absent today.

If Disha is absent today then not p can be stated as false. If Disha is not absent today then not p can be said to be true.


Conjunction

The conjunction can be said to be a compound statement which represents the word 'and'. Let us take the examples of two statements.

p = Disha is absent today.

q = The exam had begun.

What will be the conjunction of p and q? The conjunction will be, " Disha is absent today, and the exam has begun. We can say this statement to be true only if both p and q can be said to be true. If p or q is false, then the conjunction will also become false.


Truth Tables and Logical Statements 

A Truth table mainly summarizes truth values of the derived statement for all possible combinations in Boolean algebra. Truth values are the statements that can either be true or false and often represented by symbols T and F. Another way of representation of the true value is 0 and 1.


Truth Table Operations

The most accepted way of representation of a statement is through small letters such as p, q, r. There are two types of statement, i.e., open and compound statement

Open Statement: Contains one or more than one variable. So, when they are assigned specific values, it becomes a statement.

Compound Statement: When two or more simple statements are combined with words such as ‘or’, ‘and’, ‘if’ and ‘only if,’ the resultant statement is known as a closed statement. 

 

Truth Table Construction Methodology 

A. For Unary Operations

A table is prepared for a compound statement with rows and columns, and the rows quantity depends on the number of statements. The top column denotes the sub-statements variable. In the proceeding columns, you have to write truth values, and the last column is for the truth values of the compound statement. As per general rule, if there are no sub-statements, then the truth table will have 2n rows. Here, we have mentioned the truth table examples:

Case 1: Logical Truth Table: There will be a return or output to every input

p

T(p)

T

T

F

T

Case 2: Logical False’s Truth Table: False return or output to every input

p

T(p)

T

F

F

F

Case 3: Negation Truth Table: In this, the return will be the opposite of the input truth value. It denotes the truth table for NOT. You will get the opposite value of the proposition. For instance, if the incoming value is true, then the outcoming value is false.

P

~p

T

F

F

T

T

F

B. For Binary Operations

In binary operation, there are two input variables. Here, we have enlisted a few binary operations in the image format:-

Operation 

Symbol

Representation

Addition

+

a + b 

Subtraction 

-

a - b

Multiplication 

× or .

a × b or a . b

Division 

÷

a ÷ b

C. AND Operations

It is also named as a conjunction truth table. When two simple sentences p and q are connected through ‘and’, the compound statement formed is a conjunction logical table. In the case of AND operation, the output variable is true only if both input variables are true. It is denoted as p^q or p AND q. Its truth table example is given below: 

p

q

p^q

T

T

T

T

F

F

F

T

F

F

F

F

D. OR Operations

You may call it a disjunction logical table. For the feasibility of OR Operation, at least one input operand should be true. In this operation, two simple statements are linked by ‘or’ connective. It is represented as p˅q. For practice, check more disjunction truth tables examples and answers PDF.

p

q

p˅q

T

T

T

T

F

T

F

T

T

F

F

F

E. Conditional/Implication Operations

This operation is linked with the condition and two simple statements p and q are joined by the phrase ‘if and then’. The truth table for implication logical operation p→q is given below:

p

q

p→q

T

T

T

T

F

F

F

T

T

F

F

T