[Maths Class Notes] on Tautology Pdf for Exam

What is Tautology in Math?

Mathematics is a practical subject that has several concepts. Getting straight into the discussion question of what is tautology can be nerve-wracking for a beginner. There are different ways to explain what is a tautology as the Tautology definition has been represented in various forms across various course modules. But the core remains the same. As per the standard mathematical format, tautology meaning is encapsulated in the following idea. It is a sentence (or a statement in the mathematical world) that is always true. A tautological statement can never be false. It is of the utmost importance when it becomes necessary to have the truest answers or outcomes. 

Tautology Math 

Use of tautology in Math is carried out to determine that the obtained answers are absolutely true and accurate. As per the actual tautology definition, there are two forms of explanation for tautology meaning.

Tautology in Math or in logic is a statement that will always be true or will always give the answer as true. Tautology in literal sense refers to different words or a collection of words used to express the same thought or views.

Tautology Logic

Mathematics being a logical subject uses a number of logical statements before an answer can be determined. Tautology Logic also hinges on the practical reasoning that is analysed as per the set guidelines or pre-defined rules.

In order to determine whether a given statement is tautological or not, the core tautology logic must hold true. There are a number of procedures or methods are carried out by using the logical operators through which you can ascertain whether the tautology logic holds true or not. If the tautology logic holds true, then the given statement is a tautology.

Consider A Table of Two Input Values P and Q as Follows.

 The True and False values of the inputs are represented by T and F respectively. When logical operations are applied to them, they would give output according to the operator being used.

The outputs of the logical operators, which helps in practical understanding of tautology definition are as follows. 

• AND: The output is ‘True’ only when both the input values are True. 

• OR: The output is ‘True’ when either of the input values is True. 

• NOT: The output is ‘True’ when the value is False. 

• CONDITIONAL: The output is False when the first input value is ‘True’ and second input value is False. For the rest of the input combinations, the output is True.

• BICONDITIONAL: The output is True only if both the input values are True or if both the input values are False. For either of the input values being True or False, the value is False.

Tautology Truth Table

Truth tables are a graphical representation of rows and columns that show the variations of the truth or falsity of a made proposition. They represent all the possible outcomes that a problem statement can have.

The conditions from the mathematical statements are considered to be the truth values. The values in the columns are matched and the output is declared accordingly.

Consider an example for a better understanding of ‘what is tautology’.

Here, p and q are the input statements. p^q shows the ‘AND’ operation performed on the statements. (p^q) → p is a compound statement that consists of the individual p and q statements.

From the above table, it can be seen that the output for the compound statement (p^q) → p is ‘True’ in spite of the values of either of the individual statements being false.

This shows that the compound statement (p^q) → p is a Tautology. Let us look at some of the tautology examples.

Solved Tautology Examples

Example 1. Look at the figure given below. Identify if [(p→q)^p]→p is a Tautology or not.

Answer: There are 2 individual statements p. The Conditional operator is used on both of these statements. It is shown by the equation p→q. The ‘AND’ operator is then applied on the conditional statement of p→q which gives a compound statement with the equation (p→q)^p.

When a conditional operator is applied on equation (p→q)^p as well, a final compound statement is obtained which has the equation [(p→q)^p]→p.

Now, the output values obtained for the final compound statement [(p→q)^p]→p are ‘’True’’ in spite of the outputs of the previous compound equations or the individual sentences being False. This shows Tautology. This was the first of the two tautology examples, now we suggest you solve a similar question on tautology for better understanding. 

Hence, as the truth values of  [(p→q)^p]→p are {T, T, T, T} it is a Tautology.

Example-2. Prove that (P → Q) ∨ (Q → P) is a tautology or not?

Now, to prove whether the given statement is tautology or not we have to draw a truth table. (see below)

Now, as we can see the 5th column only has T, hence the given statement is a tautology. After a couple of  tautology examples, we will see some FAQs.