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A tautology is a compound statement that is true for every value of the individual statements.
- The word Tautology is derived from the Greek words tauto and logy.
- The word ‘tauto’ means ‘same’ and ‘logy’ means ‘science’.
- Tautology meaning is encapsulated in the following idea that a tautological statement can never be false.
- It is the most important part when we have to find the truest answers or results.
- Tautology in mathematics is used to determine that the obtained answers are absolutely true and accurate.
| Table of Content |
Key Terms: Tautology, Logical operator, AND, OR, NOT, Conditional, Statement, Biconditional, Operators symbol, Fallacy
Tautology in Math
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A tautology is a compound statement that always gives a truth value.
- It doesn’t matter what the individual part consists of, the result in tautology is always true.
- The contradiction or fallacy is the opposite of tautology.
- In order to determine whether a given statement is tautological or not the core tautological logic must hold true.
- There are a number of procedures or methods that are carried out by using logical statements before an answer can be determined.
Example of TautologyExample: Consider
These two individual statements are connected with the logical operator “OR”. Note: The logical operator “OR” is generally denoted by “⋁”. So, we can write the above statement as P ⋁ Q. Now, we will check whether the given statement produces a valid answer. Case I: Ram will give 5 rupees. In case I, the first statement is true and the second statement is false because the given statement is connected using the OR operator. So, it results in the true statement. Case II: Ram will not give him 5 rupees. In case II, the first statement is false and the second statement is true. Thus it gives a true statement. Now, create a truth table for the given statements
The given statement is Tautology as the final column of the Truth Table is true for all the values. |
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| Maths Related Concepts | ||
|---|---|---|
| Difference Between Relation and Function | Logarithms | Derivatives |
| Geometric Mean | Logarithm Formula | Integration |
Tautology Operators
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The following are the Tautology operators
- 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 both the input values are False.
- Conditional: The output is False when the first input value is ‘True’ and the 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 going to be False.
Tautology Logic Symbols
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Tautology uses different logical symbols to present compound statements. Some tautological symbols with their meaning and representation are tabulated below:
| Symbols | Meaning | Representation |
|---|---|---|
| ⇒ | Implies or if-then | A⇒B |
| ⇔ | If and only if | A⇔B |
| ¬ | Negation | ¬A |
| ⋀ | AND | A ⋀ B |
| ⋁ | OR | A ⋁ B |
| ∼ | NOT | ∼A |
Tautology Truth Tables of Logical Symbols
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Logical symbols are used to define a compound statement which is formed by connecting the simple statements. There are five major types of operations; AND, OR, NOT, Conditional, and Biconditional.
The truth table of all the logical operations is given below.
AND Operation
AND operation is represented by the symbol ‘⋀’. Let P and Q be two statements. The table below depicts the use of the AND symbol.
| P | Q | P ⋀ Q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
OR Operation
OR operation is represented by the symbol ‘⋁’. Let P and Q be two statements. The table below depicts the use of the OR symbol.
| P | Q | P ⋁ Q |
|---|---|---|
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
NOT Operation
NOT operation is called negation. It is represented by the symbol ‘¬’. Let P be the given statement then the value of ¬P is given as:
| P | ¬P |
|---|---|
| T | F |
| F | T |
Conditional Operation
When two simple statements are connected with the phrase ‘if and then’ to form a compound statement then in that statement conditional operation is used. Conditional operation is represented by the symbol ‘⇒’. Let P and Q be two statements. The table below depicts the use of the conditional operation.
| P | Q | P ⇒ Q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
Bi-Conditional Operation
When two simple statements are connected with the phrase ‘if and only if’ to form a compound statement then in that statement Bi-conditional operation is used. Bi-conditional operation is represented by the symbol ‘⇔’. Let P and Q are two statements. The table below depicts the use of the bi-conditional operation.
| P | Q | P ⇔ Q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |
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Things to Remember
- Tautology is a logical compound statement that at the end gives you the result as true regardless of individual statements.
- The opposite of tautology is called Fallacy or Contradiction in which the compound statement is always false.
- Logic and its representatives are very important in tautology so remember them accordingly.
- 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.
- When two simple statements are connected with the phrase ‘if and then’ to form a compound statement then in that statement conditional operation is used. Conditional operation is represented by the symbol ‘⇒’.
- When a compound statement is formed by two simple statements connected with the phrase ‘if and only if’ then the Bi-conditional operation is used. Bi-conditional operation is represented by the symbol ‘⇔’.
Sample Questions
Ques. What is tautology meaning in maths? (1 Mark)
Ans. Tautology meaning is quite easy which helps us to find the meaning of complex statements in Mathematics.
Ques. What is the difference between tautology and fallacy? (2 Marks)
Ans. A tautology is a logical compound statement that always produces the truth (true value). Fallacy or contradiction is the inverse of tautology, in which the compound statements are always false.
Ques. How to find the tautology of the given statement? (2 Marks)
Ans. The truth table makes it simple to find the tautology of the given compound statement. If all of the values in a truth table's last column are true (T), then the given compound statement is a tautology. It is not a tautology if any of the values in the final column is false (F).
Ques. Is ¬h ⇒ h is a tautology? (2 Marks)
Ans. Given that:
- h is a statement.
- ¬h is the negation
| 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.
Ques. Show that p ⇒ (p⋁q) is a tautology. (3 Marks)
Ans. The truth table of the given statement is given below:
| p | q | p ⋁ q | P ⇒ (p⋁q) |
|---|---|---|---|
| T | T | T | T |
| T | F | T | T |
| F | T | T | T |
| F | F | F | T |
So, from the result of the final column we can say that it is a tautology.
Ques. Prove that the statement (p ⇒ q)⇔(~q ⇒ ~p) is a tautology. (3 Marks)
Ans. First of all, make a truth table of the given problem.
| p | q | p ⇒ q | ~q | ~p | ~q ⇒ ~p | (p ⇒ q)⇔ (~q ⇒ ~p) |
|---|---|---|---|---|---|---|
| T | T | T | F | F | T | T |
| T | F | F | T | F | F | T |
| F | T | T | F | T | T | T |
| F | F | T | T | T | T | T |
Ques. Is x ⇒ (x⋁y) a tautology? (3 Marks)
Ans. Let’s make a truth table for this problem.
| x | y | x ⋁ y | x ⇒ (x⋁y) |
|---|---|---|---|
| T | T | T | T |
| T | F | T | T |
| F | T | T | T |
| F | F | F | T |
Since the values of the last column always stay true, so the given statement is Tautology.
Ques. Is (p ⋁ q) ⇒ (p ⋀ q) a tautology? (3 Marks)
Ans. Let’s make a truth table of the given problem,
| p | q | (p ⋁ q) | (p ⋀ q) | (p ⋁ q) ⇒ (p ⋀ q) |
|---|---|---|---|---|
| T | T | T | T | T |
| T | F | T | F | F |
| F | T | T | F | F |
| F | F | F | F | T |
Since the last column doesn’t stay true all the time. So, the given statement is not a tautology.
Ques. Is (r ⇒ s)⇔ (s ⇒ r) a tautology? (3 Marks)
Ans. The truth table of the given problem is:
| r | s | r ⇒ s | s ⇒ r | (r ⇒ s)⇔ (s ⇒ r) |
|---|---|---|---|---|
| T | T | T | T | T |
| T | F | F | T | F |
| F | T | T | F | F |
| F | F | T | T | T |
Since the last column doesn’t stay true all the time. So, the given statement is not a Tautology.
Ques. Is (p⋀q) ⇒ p a tautology? (3 Marks)
Ans. The truth table of the given statement:
| p | q | P⋀q | (p⋀q) ⇒ p |
|---|---|---|---|
| T` | T | T | T |
| T | F | F | T |
| F | T | F | T |
| F | F | F | T |
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