Now, we want to define what we call as logically equivalent statement. So before trying to understand what are logical equivalent statements? Remember in algebra and in mathematics, you often come across expressions of this form. We say for instance that a^2 + 2ab + b^2 is equal to (a + b)^2. That means these two expressions are the same expression. What do I mean by same expression? Well, by that I mean that whatever value you assign to a and b, the left hand side and right hand side will give you the same answer. That is why these two expressions are the same expression. In the same way in mathematical logic if we have a compound proposition X and a compound proposition Y then I say that they are logically equivalent and I use this notation ≡. This is not an “equal to” notation, this is representation of equivalence, this is also called as an equivalence notation. So I say that X and Y are logically equivalent if they have the same truth values. What I mean by that is I mean that if X is true then Y is true if X is false then Y is false that means it never happens that X and Y takes different truth values.

We will come back to that point later but let me first define a bi conditional operator or a bi conditional statement which for which we use this notation ↔ that means an arrowhead which has an arrowhead at both ends. And this bi conditional statement is used to represent statements of the form p if and only if q or in short form p if and only if q says another way another form of representing if and only if is iff. So very often for mathematical and for various theorem statements, you must have seen conditions like prove that this is true if and only if this holds right? So wherever we are making statements of that form, we are actually making statements of the form p bi-implication q.

Now let us next define tautology, contradiction and contingency. So a tautology is a proposition which is always true, irrespective of what truth value you assigned to the underlying variables. So, for example, if I consider this compound statement namely the disjunction of p and ¬ p, then this will be always true; that means if p is true, then this is true and even for p equal to false this statement is again true. That means it does not matter whether your p is true or false; this statement is disjunction of p and ¬ p will always be true and hence this is a tautology. Whereas a proposition is called a contradiction if it is always false irrespective of what truth value I assign to the underlying variables. So an example of contradiction is p conjunction ¬p. So you can verify that if p is false then this statement is false. And even for p equal to true this overall statement is false that means this statement is always false for every possible truth assignment of p and hence it is a contradiction. Whereas a contingency is a proposition, which is neither a tautology nor a contradiction that means it can be sometimes true it can be sometimes false. I cannot say that it is always true or it is always false.

So there are various standard logical equivalent statements which are available which are very commonly used in mathematical logic and they are also called by various names. So for instance, the conjunction of p and true is always p, that is called this law is called as the identity law. In the same way we have this double negation law which says that if you take the negation of negation of p, then that is logically equivalent to p. We have this De Morgan’s law which is very important which says that if you have a negation outside then you can take the negation inside and split it across the various variables and if you have conjunction inside then it becomes disjunction and vice versa. We also have this distributive law this says that you can distribute the disjunction over conjunction and so on.

How do we verify whether these logical identities are correct? Well, we can verify using the truth table method namely we can draw, we can construct a truth table of the left hand side of the expression, we draw the truth table of the right hand side of the expression and verify whether the truth tables are the same. So for instance, if you want to verify the De Morgan’s law, so the first part of the De Morgan’s law says that the negation of conjunction of p and q is logically equivalent to negation p disjunction negation q.

However, the truth table method of verifying logically equivalent statements has a limitation. Namely, the limitation here is it works as long as the number of variables the number of propositional variables which are there in your identity or the statement this is small. So in all this logical identity that I have written down in this table, there are at most three propositional variables and if I try to draw the truth table of a statement having 3 variables, and there will be only 8 rows which are easy to manage. But imagine I have a logical identity which has a 20 number of variables then the number of rows and that truth table will be 2^20 and definitely you cannot draw such a huge table.

So that is why it is infeasible to verify the logical equivalence of statements using the truth table method and that is why what we do here is we use some standard logical equivalent statements. So for instance, these are some of the standard logical equivalent statements, which we use to simplify complex expressions and verify whether those complex expressions are logically equivalent or not.

So now let us do an example here. Suppose I want to prove that my LHS expression and RHS expression, they are logically equivalent so this is my statement X this is my statement Y. Well in this case I can use the truth table method because my expressions X and Y involve only 2 variables and I can draw truth table which will have only 4 rows, but what I want to do here, I want to demonstrate here is that without even drawing the truth table, I can show that the expression X is logically equivalent to expression Y by using logical identities.