Tautology
A tautology is a compound statement which is true for every value of the individual statements. The word tautology is derived from a Greek word where ‘tauto’ means ‘same’ and ‘logy’ means ‘logic’. A compound statement is made with two more simple statements by using some conditional words such as ‘and’, ‘or’, ‘not’, ‘if’, ‘then’, and ‘if and only if’. For example for any two given statements such as x and y, (x ⇒ y) ∨ (y ⇒ x) is a tautology.
The simple examples of tautology are;
- Either Mohan will go home or Mohan will not go home.
- He is healthy or he is not healthy
- A number is odd or a number is not odd.
Table of contents:
- Logic Symbols
- Comparison with contradiction
- Truth tables
- Definition in Math
- Examples
- Practice problems
- FAQs
Tautology in Math
A tautology is a compound statement in Maths which always results in Truth value. It doesn’t matter what the individual part consists of, the result in tautology is always true. The opposite of tautology is contradiction or fallacy which we will learn here. It is easy to translate the tautologies from the ordinary language to mathematical expressions with the help of logical symbols. For example, I will give you 10 rupees or I will not give you 10 rupees.
Here, let us take:
P = I will give you 10 rupees
~P = I will not give you 10 rupees (Since it is the opposite statement of P)
These two individual statements are connected using the logical operator “OR” which is generally denoted by the symbol “∨”.
Thus, the above-given statement can be written as P ∨ ~P.
Now, we will check whether the given statement produces a valid answer.
Case 1: I will give 10 Rupees. In this case, the first statement is true and the second statement is false. As the given statement is connected using the OR operator, it results in the true statement.
Case 2: I will not give 10 Rupees. In this case, the first statement is false and the second statement is true. Thus, it produces a true statement.
Now, let us discuss this statement with the help of 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 |
Thus, the final column of the truth table is true for all the values, hence the given statement is a tautology.
Tautology Logic Symbols
Tautology uses different logical symbols to present compound statements. Here are the symbols and their meaning used in Maths logic:
| 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 have already discussed the term tautology, which is true for every value of the two or more given statements. The contradiction is just the opposite of tautology. When a compound statement formed by two simple given statements by performing some logical operations on them, gives the false value only is called a contradiction or in different terms, it is called a fallacy. If (x ⇒ y) ∨ (y ⇒ x) is a tautology, then ~(x ⇒ y) ∨ (y ⇒ x) is a fallacy/contradiction.
Also, read:
- Statements In Mathematical Reasoning
- Use Of If Then Statements In Mathematical Reasoning
- Mathematical Reasoning With Examples
- Important Questions Class 11 Maths Chapter 14 Mathematical Reasoning
We will also create a truth table here for better understanding the tautology and contradiction, but before that let us learn about the logical operations performed on given statements.
Tautology Truth Tables
Logical Symbols are used to connect to simple statements, to define a compound statement and this process is called as logical operations. There are 5 major logical operations performed on the basis of respective symbols, such as AND, OR, NOT, Conditional and Bi-conditional. Let us learn one by one all the symbols with their meaning and operation with the help of truth tables.
AND Operation
AND is represented as ‘∧’ symbol. When two simple statements are used to form a compound statement using AND symbol, then it is called a conjunction of two statements.
Let x and y are two statements. See the table below to perform operation using AND symbol.
| 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 |
NOT Operation
When the truth value of a statement is changed using the word NOT, it is called as a negation of the given statement. It is denoted by ‘~’ symbol. Let x 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
When a compound statement is formed by two simple statements, connected with the phrase ‘if and only if’, that is called bi-conditional operation, where the bi-conditional symbol is denoted by ‘⇔’. It also indicated as an equivalent symbol.
| x | y | x ⇔ y |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |
Tautology Definition in Math
Let x and y are two given statements. As per the definition of tautology, the compound statement should be true for every value.
The truth table helps to understand the definition of tautology in a better way. Now, let us discuss how to construct the truth table. Generally, the truth table helps to test various logical statements and compound statements. The first part of the compound statement is symbolized in the first column of the truth table. The second part of the compound statement, which is following the logical connector, is symbolized in the second column. The logical connectors such as and, or, etc provide the meaning of the compound statement. The third column of the truth table should contain the relationship between the two statements. If every result in the third column is True (T), then the given compound statement is a tautology.
| x | y | x ⇒ y | y ⇒ x | Tautology = (x ⇒ y) ∨ (y ⇒ x) | Contradiction = ~(x ⇒ y) ∨ (y ⇒ x) |
| T | T | T | T | T | F |
| T | F | F | T | T | F |
| F | T | T | F | T | F |
| F | F | T | T | T | F |
Tautology Examples
Example 1: Is ~h ⇒h is a tautology?
Solution: Given ‘h’ is a statement.
| h | ~h | ~h ⇒h |
| T | F | T |
| F | T | F |
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:
| p | q | p∨q | p⇒(p∨q) |
| T | T | T | T |
| T | F | T | T |
| F | T | T | T |
| F | F | F | T |
The truth values of p⇒(p∨q) is true for all the value of individual statements. Therefore, it is a tautology.
Example 3: Find if ~A∧B ⇒ ~(A∨B) is a tautology or not.
Solution: Given A and B are two statements. Therefore, we can write the truth table for the given statements as;
| A | ~A | B | ~A∧B | A∨B | ~(A∨B) | ~A∧B ⇒ ~(A∨B) |
| T | F | T | F | T | F | T |
| T | F | F | F | T | F | T |
| F | T | T | T | T | F | F |
| F | T | F | F | F | T | T |
Hence, you can see from the above truth table ~A∧B ⇒ ~(A∨B) is not true for all the individual statements. Therefore, it is not a tautology.
Practice Problems
Check that the following statements are tautology or not.
- p ∨ ¬p
- p ∧ ¬p
- q → (p ∨ q)
- (p ∨ q) ∧ (¬p) ∧ (¬q)
- (p ∧ q) → p
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Frequently Asked Questions on Tautology
What is meant by tautology in Maths?
In Mathematics, a tautology is a logical compound statement, which results in a true statement, regardless of individual statements.
What is the difference between tautology and fallacy?
If a logical compound statement always produces the truth (true value), then it is called a tautology. The opposite of tautology is called fallacy or contradiction, in which the compound statement is always false.
How to find the tautology of the given statement?
The tautology of the given compound statement can be easily found with the help of the truth table. If all the values in the final column of a truth table are true (T), then the given compound statement is a tautology. If any of the values in the final column is false (F), then it is not a tautology.
What does A∨B mean in logic?
In a logical statement, A ∨ B is a compound statement, which is connected using the “OR” operator. Here, A and B are the individual statements. In OR operation, if any one of the statements holds true value, then the compound statement results in true value. If both the statements are false, then the compound statement is also false.
What are the symbols used in tautology?
The important logic symbol used in tautology are:
AND (∧)
OR (∨)
NOT (~)
Negation (¬)
Implies (→)
If and only if (⇔)