Tautology
TAUTOLOGIES
Tautology is defined as, a statement formula which is true regardless of the truth values of the statements which replaces the values variables in it is called a universally valid formula or a tautology or a logically truth.
In generally tautology is defined as a statement formula of truth table consists of all truth values in the result column. We may say that a statement formula which a tautology is identically true.In other words, a statement formula which is false regardless of the truth values of the statements which replaces the variables in it is called a contradiction. Obviously , the negation of a contradiction is a tautology. A formula which is contradiction id identically false.
In general method to determine whether a given formula is a tautology is to construct its truth table.This process can always be used but often becomes tedious, particularly when the number of distinct variables is large are when the formula is complicated. “A simple fact about tautologies is that the conjunction of two tautologies is also a tautology”. Let us denote by A and B are statements formulas which are tautologies if we assign any truth values to the variables of A and B , then the truth values of both A and B will be True. Thus the truth values of A∧ B will be True, so that A∨B will be tautology.” The importance of the above concept lies infact that any substitution instance of a tautology is a tautology”.
Some problems on tautology :
1.(A→B) ⇔ (~A∨B)
A  B  ~A  ~A∨B  (A→B)  (A→B) ⇔ (~A∨B) 
T  T  F  T  T  T 
T  F  F  F  F  T 
F  T  T  T  T  T 
F  F  T  T  T  T 
2.((P→Q) ∧ (Q→R)) → (P→R)
P  Q  ~P  ~Q  P∨ Q  ~P∨Q  (P∨ Q) ∨ (~P∨ Q)

~((P∨ Q) ∨(~P∨Q))

T  T  F  F  T  T  T  F 
T  F  F  T  T  F  T  F 
F  T  T  F  T  T  T  F 
F  F  T  T  F  T  T  F 
Some problems on contradiction:
1.~(P→Q) ∨ (Q→P)
P  Q  P→Q  Q→P  (P→Q)∨(Q→P)

~(P→Q) ∨(Q→P)

T  T  T  T  T  F 
T  F  F  T  T  F 
F  T  T  F  T  F 
F  F  T  T  T  F 
2.~((P∨ Q) ∨ (~P∨ Q))
P  Q  ~P  ~Q  P∨Q  ~P∨Q  (P∨Q)∨(~P∨Q)

(~((P∨Q)∨(~P∨Q)))

T  T  F  F  T  T  T  F 
T  F  F  T  T  F  T  F 
F  T  T  F  T  T  T  F 
F  F  T  T  F  T  T  F 
Leave a reply