In this lecture, we are going to talk about the logical equivalences. Now, let's consider the definition of logical equivalence. The compound propositions P and Q are said to be logically equivalent if P biconditional operator Q is a tautology. Now, what does it really mean? When P is true, then Q has to be true and when P is false, then Q has to be false.
then only it will be a tautology. Right? Logical equivalence is denoted by this sign or this sign.
Let's consider one example. P and true is equivalent to P. How we can say that P and true is equivalent to P? Let's try to understand with the help of the truth table. Let P be the preposition.
There are two possible truth values for this preposition that is true or false. Now, T means true. That means in all the cases it will be true, right?
And p and true would be true and true is true, false and true is false. Now you can see that p and true is equivalent to p. The truth values are same.
Right? When p is true, then p and true is true. And when p is false, then p and true is also false. Therefore we can say that p and true is equivalent to p. Right?
Now let's consider the most common. and famous logical equivalences of all time. The first one is identity laws. P and true is equivalent to P which we already see in this example. P or false is equivalent to P.
You can logically think about this statement. Why P or false is equivalent to P? We know that if P is true, then true or false will be true which is equivalent to P. And when p is false, then false or false is equivalent to false. Then also it is equivalent to p.
That is why we can say that p or false is equivalent to p. Right? The second one is domination laws.
p or true is equivalent to true. Now this is very simple to understand. As we know according to disjunction, if at least one of the prepositions is true, then the whole compound preposition will be true.
And this is what we got as the result. P and false is equivalent to false. Now, this is also true. According to conjunction operator, if any one of the proposition is false, then the whole compound proposition will be false.
And that is what we are getting from this statement. The third one is idempotent laws. P or P is equivalent to P. P and P is equivalent to P. This can also be very easily understood, right?
We can always check this with the help of the truth table. Fourth one is double negation law. Not, not p is equivalent to p. It's just like both nots get cancelled out and the result we got is p.
Fifth one is commutative laws which is very simple to understand. p or q is equivalent to q or p. p and q is equivalent to q and p. It doesn't matter what is the order of the preposition.
The final result of both of them will be same. The sixth one is associative laws. p or q or p is equivalent to q or p. or R is equivalent to P or Q or R. Here it doesn't matter which one is calculated first.
In a compound preposition P or Q or R, which one is calculated first, it doesn't really matter. This is what associative laws is all about. Either P or Q will be calculated first or may be Q or R will be calculated first.
It doesn't really matter. Similarly for the conjunction operator as well, this is satisfied. Now the seventh one is distributive laws. P or Q and R is equivalent to P or Q and P or R.
P or Q and R, it distributes along P or Q and P or R. We can always check this with the help of the truth table. That they both are equivalent. Now, this is also applicable if we change OR with AND and AND with OR. Like in this case, P and and q or r is equivalent to p and q or p and r.
Okay. Eighth one is De Morgan's laws. Not of p and q, this not will go inside, change the conjunction to disjunction and negate both these prepositions. This will be equivalent to not p or not q.
Similarly, this will happen for the disjunction as well. This negation negates both the propositions as well as change this disjunction to conjunction. That is, not p and not q.
Ninth is absorption laws. That is P or P and Q is equivalent to P. P and P or Q is equivalent to P.
Now this is little bit difficult to understand. The point here is that, it seems like a distributive law, right? P or P and Q, like here it is P or Q and R. Here in case of distributive laws, we are having three prepositions P, Q and R. But in case of absorption laws, we are having only two prepositions P and Q.
This is the only difference. Now, if we try to solve this with the help of distributive laws, we will have. Obviously, we can check this with the help of truth table also that p or p and q is equivalent to p. p or p and q is equivalent to p while p and p or q is equivalent to p as well.
Now, the last one is negation laws. p or not p is equivalent to true p and not p is equivalent to false. Why? We already considered this as an example in the previous lecture that p or not p is a tautology while p and not p is a contradiction. Right?
After considering all these famous logical equivalences, it's time to consider logical equivalences involving conditional statements. Let's consider them one by one. p implies q is equivalent to not p or q. Now this is very interesting rule and one of the most important rules of all time. This is very very important.
We can think of this logically that p implies q is equivalent to not p or q. We already know that a compound preposition involving disjunction is always true when at least one preposition in that compound preposition is true. Let's say not p is true.
If not p is true, then we know that p would be false. And we already know that when p is false, then p implies q would always be true. Here also not p or q is true because not p is true.
Let's say suppose q is true. Now if q is true, then this q is also true. Now we know that in p implies q, if q is true, then p implies q would always be true.
And hence again both are equivalent. Now, let's try to prove that when not P or Q is false, then P implies Q would also be false. This can be false when both of them are false. That is, when not P is false and Q is also false. If not P is false, then P would be true.
And if Q is false, then here also Q is false. True implies false is always false in case of implication. And here also, false or false is false.
Again, both of them are equivalent. When this is true, then this is also true. When this is false, then this is also false.
And hence, P implies Q is equivalent to not P or Q. Right? And it is hence proved. Let's consider the second logical equivalence.
P implies Q is equivalent to not Q implies not P. As not Q implies not P is the contrapositive of of p implies q. Hence, they both are equivalent. And we already learnt that they both are equivalent.
And hence, p implies q is equivalent to not q implies not p. p or q is equivalent to not p implies q. Let's prove this one. As we know p implies q is equivalent to not p or q. We are going to use this particular logical equivalence here. Can we replace not p implies q with not of not p or Q.
How can we write like this? As we know P implies Q is equivalent to not P or Q. Here instead of P, we are having not P. Hence, we are going to replace this P with not P here.
This will become not of not P or Q. According to the double negation law, this will get cancelled out and the final result would be P or Q. Right?
Which is this one. Hence we can say not P implies Q is equivalent to P or Q. Okay. Now the next one is P and Q is equivalent to not of Q implies not P.
This is also very interesting law. Let's prove this one as well. Not of Q implies not P. We can write Q implies not P as not Q or not P. Right?
According to this particular rule. According to De Morgan's law, when this negation goes inside, it will change the signs. Not Q will become Q, Not P will become P, disjunction will become conjunction.
Right? This is Q and P. Now, according to commutative law, this will become P and Q.
Right? Q and P is equivalent to P and Q. And this is what we are getting as a result. P and Q.
Let's consider the next rule. Not of P implies Q is equivalent to P and Not Q. We can easily prove this as well. This will be your homework. Okay? Sixth one is P implies Q and P implies R.
This is equivalent to P implies Q and R. How is it like that? Let's try to prove this one. P implies Q and P implies R.
We can write this P implies Q as not of P or Q. Again according to the first rule. And P implies R as not of P. or R.
Now, according to the distributive rule, this is not of P or Q and R. Apply the distributive law here and you would be able to obtain this particular expression. This will be equivalent to not of P or Q and not of P or R. Now, let's rename this expression Q and R as S. Then this will be not of P or S.
Right? Not of P or S can be rewritten as P implies S. Right? What is S?
S is Q and R. And this is what we are getting as a result. P implies Q and R. Right?
Similarly, we can prove this that P implies R and Q implies R is equivalent to P or Q implies R. P implies Q or P implies R is equivalent to P implies Q or R. P implies R or Q implies R is equivalent to P and Q implies R.
Now, let's consider the logical equivalences involving biconditional statements. We know that P biconditional operator Q is equivalent to P implies Q and Q implies P. P is not only sufficient condition for Q, but P is also a necessary condition for Q. Right? Now the second logical equivalence would be P biconditional operator Q is equivalent to not P biconditional operator not Q.
Let's rewrite this statement as P implies Q and Q implies P. Now, P implies Q can be rewritten as not Q implies not P. This is the contrapositive of P implies Q. They both are equivalent, right? So, we can replace this P implies Q with not Q implies not P.
And we can similarly replace Q implies P with not P implies not Q. According to the commutative law, we can write this particular expression first. and this one last. This will be not p implies not q and not q implies not p.
Let's say this one is s and this one is r. Then we can rewrite the statement as s implies r and r implies s. Which is equivalent to s biconditional operator R. As S is not P, so we can write it as not P biconditional operator R is not Q. So, we can write not Q instead of R.
And this is our final result. Now, let's consider the third logical equivalence. P biconditional operator Q is equivalent to P and Q or not P and not Q. Now, this is also very simple to prove.
as p by conditional operator q is equivalent to p implies q and q implies p. Now, we can write p implies q as not of p or q and q implies p as not of q or p. Now, we know if we have an expression like a plus b C plus D. We expand it like AC plus AD plus BC plus BD.
We multiply A with C, we multiply A with D, we multiply B with C, we multiply B with D. That's what we do when we have an expression like this. This expression is also equivalent to this particular expression.
Here, this is representing the OR and this is representing the AND. Now, let's expand this particular equation as well. This will be not p and not q or not p and p or q and not q or finally q and p.
Not p and p is false. Right? Similarly, q and not q is also false according to the negation laws.
Then this expression would be not p and not q or false or false or q and p. As false or with anything will be anything. Right?
So, this is equivalent to not P and not Q or Q and P. According to the commutative law, we can write Q and P as P and Q. Not P and not Q or P and Q.
And this is what we are getting finally. Right? Now let's consider the fourth expression.
Not of P by conditional operator Q. is equivalent to p biconditional operator not q. Now, this logical equivalence can also be proved very easily using the laws that we already learnt.
So, this will be your homework.