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:

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.

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 .

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