In the previous post , we learned about the prepositional statements and truth tables. You can visit the first post by clicking here.

**Tautology**

Formal definition of Tautology, “**It is a statement which is True for all it’s variable values”.**

Meaning, It is a compound statement and each of it’s variable has a truth value and combination of those truth value give an output which is always **true**. It’s very easy to understand using a truth table.

#### example:

Truth Table for P v P’ |

We can see that the output of p v p’ is True for all combination. This is called a **Tautology**.

**Contradiction**

Formally, “**A Contradiction is a statement which is false for all possible assignments to is prepositional variables”.**

If a compound statement has a truth value assigned to it’s variables and there combination is always false for all possible assignments then it is a **contradiction. **We will again consider an example truth table, so that it will be easier to understand.

Truth Table for p and p’ |

Again we see that the output gives us False for all possible assignments.

**Contingency**

**“A Logical statement which is not Tautology, nor Contradiction , is called Contingency”.**

** **To understand contingency we will construct truth table for

**(p ∧ q)-> r .**

Truth Table for Contingency |

**Equivalence of Statements**

Some compound statements have same truth values , it is denoted as p <=> q.

**Dualilty Law **

It is very simple to find, we complement the connectives ∨, ∧ , ¬ and obtain the dual of a priposition p which denoted as p*.

#### example.

**s = (p ∨ q) ∨ r , then Dual is s* = p ∧ ( q ∧ r)**

**s = p v (q ∧ r), then the Dual is s* = p ∧ ( q ∨ r)**

**Problems**

**Q1.**

**¬p ∧ q**

TRUTH TABLE FOR P’ AND Q |

**Q2.**

**¬p ∧**

**(¬p**

**∧**

**q )****¬**TRUTH TABLE FOR ¬p ∧ (¬p ∧ ¬q ) |

### Exercise