Home | | Maths 12th Std | Mathematical Logic: Tautology, Contradiction, and Contingency

# Mathematical Logic: Tautology, Contradiction, and Contingency

A statement is said to be a tautology if its truth value is always T irrespective of the truth values of its component statements. It is denoted by T.

Mathematical Logic

### Definition 12.16

A statement is said to be a tautology if its truth value is always T irrespective of the truth values of its component statements. It is denoted by T.

### Definition 12.17

A statement is said to be a contradiction if its truth value is always F irrespective of the truth values of its component statements. It is denoted by F.

### Definition 12.18

A statement which is neither a tautology nor a contradiction is called contingency

### Observations

1. For a tautology, all the entries in the column corresponding to the statement formula will contain T.

2. For a contradiction, all the entries in the column corresponding to the statement formula will contain F.

3. The negation of a tautology is a contradiction and the negation of a contradiction is a tautology.

4. The disjunction of a statement with its negation is a tautology and the conjunction of a statement with its negation is a contradiction. That is p ¬p is a tautology and p ¬p is a contradiction. This can be easily seen by constructing their truth tables as given below.

### Example for tautology Since the last column of p ¬p contains only T, p ¬p is a tautology. Since the last column contains only F, p ¬p is a contradiction.

### Note

All the entries in the last column of Table 12.10 are F and hence ( p q) ( p ¬q) is a contradiction.

### Example for contingency In the above truth table, the entries in the last column are a combination of T and F. The given statement is neither a tautology nor a contradiction. It is a contingency.

Tags : Discrete Mathematics | Mathematics , 12th Maths : UNIT 12 : Discrete Mathematics
Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail
12th Maths : UNIT 12 : Discrete Mathematics : Mathematical Logic: Tautology, Contradiction, and Contingency | Discrete Mathematics | Mathematics