In the previous post , we learned about the prepositional statements and truth tables. You can visit the first post by clicking here.
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.
|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.
“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.
It is very simple to find, we complement the connectives ∨, ∧ , ¬ and obtain the dual of a priposition p which denoted as p*.
s = (p ∨ q) ∨ r , then Dual is s* = p ∧ ( q ∧ r)
s = p v (q ∧ r), then the Dual is s* = p ∧ ( q ∨ r)
|TRUTH TABLE FOR P’ AND Q|
|TRUTH TABLE FOR ¬p ∧ (¬p ∧ ¬q )|