A Tautology is a compound assertion that is true for all possible values of the separate statements. Tautology is derived from a Greek term in which ‘tauto’ means’same’ and ‘logia’ means ‘logic’. A compound statement is formed by combining two basic assertions with conditional terms such as ‘and,’ ‘or,’ ‘not,’ ‘if,’ ‘then,’ and ‘if and only if’. For any two given assertions, such as x and y, (x y) (y x) is a tautology.
Simple Tautology Examples Include:
- Mohan will either go home or will not go home.
- He is either healthy or unhealthy.
- A number is either odd or not odd.
Tautology in Math
A tautology is a compound assertion that always yields the Truth value in mathematics. The outcome in tautology is always true, regardless of what the constituent parts are. The inverse of tautology is contradiction or folly, as we will see below. With the use of logical symbols, tautologies may be easily translated from a common language to mathematical equations. For example, I will either offer you or refuse to give you ten rupees.
Consider the following:
P = I’ll pay you ten rupees.
~P = I will not give you ten rupees (Since it is the opposite statement of P)
These two separate propositions are linked by the logical operator “OR,” which is commonly represented by the symbol “∨”.
Thus, the above-given statement can be written as P ∨ ~P.
Now we’ll see if the supplied statement yields a correct response.
Case 1: I’ll give you ten rupees. In this example, the first statement is correct while the second is not. Because the provided statement is linked with the OR operator, it yields the true statement.
Case 2: I’m not going to offer you ten rupees. The first assertion is untrue in this circumstance, but the second statement is true. As a result, it yields a true assertion.
Let us now examine this assertion using the truth table.
| P ( I will give you 10 Rupees) | ~P ( I will not give you 10 Rupees) | P ∨ ~P (I will give you 10 rupees or I will not give you 10 rupees) |
| T | F | T |
| F | T | T |
As a result, the last column of the truth table is true for all values, indicating that the supplied statement is a tautology.
Tautology Definition in Math
Let x and y be two assertions. The compound assertion should be true for all values, according to the notion of tautology.
The truth table facilitates a better understanding of the definition of tautology. Let us now look at how to build the truth table. In general, the truth table aids in the testing of various logical and complex propositions. The first column of the truth table represents the first portion of the compound statement. The second column represents the second component of the compound statement, which comes after the logical connector. The meaning of the compound statement is provided by logical connectors such as and, or, and so on. The relationship between the two assertions should be mentioned in the truth table’s third column. If all of the answers in the third column are True (T), the provided compound statement is a tautology.
Tautology Logic Symbols
To convey compound claims, tautology employs a variety of logical symbols. Here are the mathematical symbols and their meanings:
| Symbols | Meaning | Representation |
| ∧ | AND | A ∧ B |
| ∨ | OR | A ∨ B |
| ¬ | Negation | ¬A |
| ~ | NOT | ~A |
| → | Implies or If-then | A→B |
| ⇔ | If and only if | A⇔B |
Tautology And Contradiction
We’ve already covered the concept of tautology, which holds true for whatever value of the two or more provided assertions. Contradiction is the polar opposite of tautology. When a compound statement generated by two simple provided assertions is assigned the incorrect value only after executing some logical operations on them, it is termed a contradiction or, in other words, a fallacy. If (x ⇒ y) ∨ (y ⇒ x) is a tautology, then ~(x ⇒ y) ∨ (y ⇒ x) is a fallacy/contradiction.
We’ll also make a truth table to help us grasp the tautology and contradiction, but first, let’s look at the logical operations that are done on provided truths.
Tautology Truth Tables
Logical symbols are used to link basic assertions in order to construct a compound statement, a process known as logical operations. AND, OR, NOT, Conditional, and Bi-conditional are the five major logical operations done on the basis of respective symbols. Let us study all the symbols, their meanings, and how they work one by one using truth tables.
AnD Operation
AND is represented by the symbol ”. A conjunction of two statements occurs when two basic assertions are utilized to produce a compound statement using the AND sign.
Consider x and y to be two propositions. To use the AND sign, go to the table below.
| x | y | x ∧ y |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
OR Operation
OR is represented by ‘∨’ symbol. When two simple statements are used to form a compound statement using an OR symbol, then it is called a disjunction of two statements.
| x | y | x ∨ y |
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
The sign ” represents OR. A disjunction of two assertions occurs when two simple statements are utilized to produce a complex statement using an OR sign.
NOT Operation
A negation of a statement occurs when the truth value of a statement is modified by employing the term NOT. It is represented by the symbol”. If we consider x to be a given statement, then x is given by;
| x | ~x (NOT x) |
| T | F |
| F | T |
Conditional Operation
When a compound statement is formed by two simple statements, connected with the phrase ‘if and then’, that is called conditional operation, where the conditional symbol is denoted by ‘⇒’. This symbol also denotes as implies.
| x | y | x ⇒ y |
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
Bi-conditional Operation
A bi-conditional operation occurs when a compound statement is generated by two simple assertions linked by the phrase ‘if and only if’, The bi-conditional symbol is indicated by ‘⇔’. It was also used as an analogous sign.
| x | y | x ⇔ y |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |
Tautology Examples
Example 1: Is ~h ⇒h is a tautology?
Solution: Given ‘h’ is a statement.
Since, the true value of ~h ⇒h is {T,F}, therefore it is not a tautology.
Example 2: Show that p⇒(p∨q) is a tautology.
Solution:
The truth values of p⇒(p∨q) are true for all individual statement values. As a result, it is a tautology.
Example 3: Find if ~A∧B ⇒ ~(A∨B) is a tautology or not.
Solution: Assume that A and B are two propositions. As a result, we can construct the truth table for the provided assertions as follows:
As you can see from the truth table above, ~A∧B ⇒ ~(A∨B) is not true for all individual claims. As a result, it is not a tautology.
Tautology : Practice Problems
Check to see if the following assertions are tautologies.
- p ∨ ¬p
- p ∧ ¬p
- q → (p ∨ q)
- (p ∨ q) ∧ (¬p) ∧ (¬q)
- (p ∧ q) → p
Frequently Asked Questions on Tautology
What exactly is a tautology in mathematics?
A tautology is a logical compound statement in mathematics that culminates in a true assertion independent of component statements.
What’s the distinction between tautology and fallacy?
A tautology is a logical compound statement that always generates the truth (true value). Fallacy or contradiction is the inverse of tautology, in which the compound assertion is always wrong.
How can I determine the tautology of a given statement?
The truth table makes it simple to find the tautology of the given compound sentence. If all of the values in a truth table’s last column are true (T), then the supplied compound statement is a tautology. It is not a tautology if any of the values in the final column is false (F).
What does A∨B stand for in logic?
A B is a compound assertion connected by the “OR” operator in a logical statement. A and B are the specific statements in this case. If any one of the claims is true in the OR operation, then the compound statement is true. If both claims are wrong, the compound statement is false as well.
What symbols are used in tautology?
The following logic symbols are commonly employed in tautology:
AND (∧)
OR (∨)
NOT (~)
Negation (¬)
Implies (→)
If and only if (⇔)
This post first appeared on RD Sharma Solutions For Class 10 Maths Chapter 1 Exercise 1.1 Real Numbers, please read the originial post: here
