 # 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.

## 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.

### 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

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.

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.

 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.

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