# Prepositional Logic and Problems – II

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.

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

Obtain truth table for the following.

Q1. ¬p ∧ q
 TRUTH TABLE FOR P’ AND Q
Q2.   ¬p ∧ (¬p  ¬q )
 TRUTH TABLE FOR  ¬p ∧ (¬p  ∧ ¬q )

### Exercise

Show that the p -> q and ¬p ∨ q both are equivalent statement.
hint : Use Truth Table to prove the logical equivalence.