In writing, statements can be evaluated in regard to one another. Depending on their relation, they may be observed as a Tautology or a Logical Equivalence.
Tautology:
If we consider a sentence, It is cool or it is not cool, it is the disjunction of a statement and its negation. These type of sentences can be represented by the disjunction pv ~p. Since the disjunction is true if at least one of its component is true, pv~q is always true because one of p and ~q is always true. The truth table for pv~q is as follows.
Here all the truth values of the compound statement pv~q are true for all possible truth values of the statement p. Here, pv~q is called a tautology.
Thus, a tautology is a compound proposition which is true for all possible combinations of truth values of its components.
Logical equivalence:
Let us consider two statements.
p: I work hard
q: I will fail
If we consider the two sentences, If I don’t work hard then I will fail and I work hard or I will fail mean the same. Let us observe the same thing symbolically with the help of truth tables.
~p ->q : If I don’t work hard then I will fail
pvq : I work hard or I will fail
Here, if we can observe that the truth values of both ~p->q and pvq are same for all possible combinations of truth values of p and q. So the two sentences are logically equivalent.
Thus, two compound statements are said to be logically equivalent if all of their truth values are same for all the possible combinations of its components.
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