 Welcome back, I am Professor R. V. R. Kiddhi from CAC department of Tupli-Tikshu-Labor. Today I will be focusing on how to solve the principal PCNF and PDNF problems that is principal conjunct normal form and principal decision to normal forms. The learning objective of today's session lecture is student can convert the given equation into principal conjunct normal form and principal decision to normal form that is PCNF and PDNF. As we know that there are possible mean terms and maximum, so two variables are four, okay. They are set A which is a max term, set B is a mean term, set A is a, okay. Then we have problem solving, okay. Let us consider the first example, okay. In this video, I am going to take two problem statements, okay, two formulas, okay, which I am going to identify, okay, the question let us first of all read the question, obtain the principal decision to and conjunct to normal forms of the following formula, okay, and also identify which of the above formula are tautologies, okay. Now one more question is added over here, earlier in the previous videos we used to find out either PDNF or PCNF, but here after finding out, after obtaining PCNF and PDNF, we are also supposed to identify whether they are having tautology or contradiction, okay. Now what is meant by tautology, what is meant by tautology, tautology is that, okay, where only you will have PDNF, principal decision to normal form, okay, you will not have PCNF, okay, that such formula is called as, such formula is called as a tautology, suppose two variables are there and it is a summation of all the four mean terms are there, okay. In a two variables we have two mean terms are there, okay, and if you get summation of all those four mean terms then that is called as a tautology, means definitely it will not have a PCNF, okay. Now this is the first problem, let us solve this particular problem, okay, the problem is negation P or negation Q conditional in the bracket P, bi-conditional, negation Q. Let us assume that this is equation number one, okay. Now as we know that we need to remove this bi-conditional first, okay, so this bi-conditional should be removed by substituting an equivalent formula, so let us see what is the equivalent formula for B, B, B bi-conditional Q, so since we know that R bi-conditional S is equal to R and S or negation R and negation S, okay, let us substitute this particular formula in the equation number one, so that equation number one will now become negation, okay, P bi-conditional negation Q is equal to it is P and negation Q or negation P and Q, okay, let us substitute this particular, this particular form, okay, into the equation number one, okay. So equation number one will become the first part of this negation P or negation Q will remain as it is, then conditional will remain as it is, for this particular P bi-conditional Q, okay, we are going to substitute these two terms that is P and negation Q or P, negation P and Q, this is equation number two, okay, now also we need to remove this conditional by substituting an equivalent formula, by substituting an equivalent formula, okay. So now this equation number two, okay, substitute or use this deep Morgan law and apply to the equation number two, so that this negation will go off, so negation will come at the end of this, at the first starting of this particular bracket, okay, so that will be negation off into bracket, negation P or negation Q or the second term or the second term P and negation Q or negation P and Q, this is equation number three. If you look at this particular formula, okay, this is R, this is R, okay, so I think still we cannot conclude that this is a PDNF, okay, this is a PDNF, we have to, what we have to do, we have to take this negation inside the bracket by using the Morgan's law, okay, so since we know that negation of negation P is equal to P, apply this equal to equation number three, okay, so this will become negation of negation equal to P, simply P and this R will change to and and negation of negation will go off and this will be remain Q, so the first term, simplification of the first term will be P and Q or P and negation Q or negation P and Q, okay, so now if you look at the equation number four, okay, it is perfectly the situation where summation of mean terms are there, okay, three mean terms are there, okay, all our mean terms are there, okay, it is summation of mean terms, definitely it is a PDNF is there, okay, it is a PDNF or it is summation of three mean terms, summation of three mean terms, so that formula will not have this one, tautology because three mean terms are there only, so here I will put a question to you, okay, it is a PDNF, what will be the PCNF, okay, the question is what is the PCNF, write down the PCNF term, okay, or the max term, okay, the max term in a PCNF for this particular given PDNF, okay, take a pause over here and try to write on a paper, piece of paper, so it is, it is answer is, answer is what is a PCNF, it is, is it P and Q or P and negation Q or negation P and Q, it is summation of, it is summation of, it is summation of three mean terms M1, M2, M3, okay, so negation S is equal to, you will get only one mean term, okay, and if you take double negation S that will be P or Q, P or Q is the required PDNF, okay, this is the required PDNF, now let us focus on the problem number two or example number two, obtain PCNF or PDNF and also identify whether it is tautology or not, okay, the question is P conditional into bracket P and Q conditional P, okay, say this is equation number one, what we will do, first of all we remove the conditional connective by substituting an equivalent formula P conditional Q is equal to negation P or Q, okay, Q conditional P is equal to negation Q or P, okay, let us substitute this particular bracket, so the equation one will become negation P or, okay, P, this conditional is also removed, that becomes negation P or into bracket P and into bracket this conditional is removed, this will become negation Q or P, so this is equation number two, so that equation number two has to be simplified now, okay, this has to be simplified, so what has to be done, negation P or, okay, P and negation Q in one bracket, okay, if you simplify this particular, so R P, okay, so this is equation number three, now this is not a PDNF, okay, because negation P is not maximum and P is also not maximum, so we need to convert that into a, sorry, mean term, we need to convert that into a mean term for the required PDNF, so use P and T is equal to P, okay, we know that P and tautology is equal to P and for tautology we are going to substitute the second variable, okay, which is missing for that particular to become a mean term, okay, now this negation P, okay, negation P and T or P and Q or P and T, okay, now in place of T we are going to substitute Q or negation Q and here also Q or negation Q will be substituted in equation number four, so this is equation number four, so put T is equal to Q or negation Q or P or negation P, okay, as per the requirement in the formula, okay, if P is there you put Q or negation Q, if Q is there you put P or negation P, now in this equation number four negation P and P are variables that are to be converted into mean terms, so only we use Q or negation Q, so once you substitute that Q or negation Q in a given equation four, okay, so equation four will become now negation P and that Q or tautology then this second term will remain as it is and third term tautology will remain this tautology is added, now use distributive law, okay, use distributive law in equation number four, so that this will become negation P and Q or negation P and negation Q or P and negation Q or P and Q, P and Q, okay, and P and negation Q, okay, if you look at this equation number six, okay, all are in summation and negation P and Q is mean term, negation P and Q is mean term, P and negation Q is mean term and P and Q is mean term, P and negation Q is mean term, five mean terms are there, okay, so we need to eliminate one of the mean term, okay, for that what we will do, we use Adam Pudett law, okay, so P and negation Q, okay, P and negation Q and P and negation Q, these are repeated, so we will write only once, okay, we will write only once P and negation Q, this P and negation Q and this third term P and negation Q, okay, we will write only once, so this is what the required PC, required PDNF, okay, so this is required PDNF, so it is summation of all the four mean terms. So this is called as a tautology also. I hope you have understood how to solve the problems of PCNF and PDNF. These are the references. Thank you very much.