 Professor R. V. R. Kiddhi from Computer Science and Engineering Department of W. T. Shalapur. Today I will be focusing on problem solving on principal conjunctive normal form. So we are going to see what are the different problems on PCNF. The learning outcome of today's lecture is you will be able to convert the given equation into principal conjunctive normal form. Let us these are the just recap the maximums okay for two variables there are four maximums they are from set A or set B okay that is P or Q negation P or Q P or negation Q negation P or negation Q okay and set B is the commutative of negation set A. So either set A or set B we need to consider. So whatever the maximums are there for two variables they are only four. So in a PCNF it will be product of only at the max four maximums or one maximums one to all four maximums are possible and we are going to identify which are those maximums from this set A that is our objective of today's session. Now the first problem statement okay the problem statement is like that write equivalent forms for the following formulas in which negations are applied to the variables only obtain the principal conjunctive normal form that is PCNF of the following formula. The formula is given in in a such a way that negation into bracket P by conditional Q okay let us see the solution. As we know that in a PCNF or in a normal form we don't have we have only basic and it is like and or a negation okay and or a negation. So in this particular formula in this particular formula in this particular formula by conditional laser so we need to remove this by conditional we need to remove this by conditional by substituting an equivalent formula and equivalent formula in a given equation. So let us say this assume that this is equation number one and since we know that P by conditional Q is equal to there is equivalent formula called as P conditional Q and Q conditional P okay let us substitute this formula in the given equation number one. So that equation number one now will become as negation into bracket okay. So it will be a negation of negation will go off okay so it will be only P conditional Q okay and Q conditional P okay that P having negation was there okay it will go off okay so it is equation number two use de Morgan's law in the equation number two okay so P conditional Q is equal to negation P or Q okay once use this particular formula in equation number two so that it will become P conditional Q will become negation P or Q and Q conditional P Q conditional P this particular formula will become negation Q or P. Now after that after that use the committed to law okay use the committed to law in equation number three that is what negation take P as a first variable and Q as a second variable okay so that it will be negation P or Q there is no change and over here in the second term it will be P or negation Q okay. So this is what the required PCNF required principle constructive noro form for the equation or for the problem statement number one let us go for the problem now I will put a question over here okay if a formula is having tautology okay if a formula is having tautology does it have PCNF and PDNF both or it has only PCNF or it has only PDNF that you need to answer take a pause in this video over here and write down the answer the answer is okay if a formula is having tautology it will have only PDNF okay it will not have a PCNF so all the mean terms will be there in a given PDNF. Now let us focus on the example number two okay the example is question is write equivalent forms for the following formulas in which negations are applied to the variables only and obtain PCNF given formula okay this is the second formula okay the formula is negation into bracket P and Q now we have to this looks like a very simple okay if you take negation insert the packet by using de Morgan's law so it will become so let us assume that is given equation is equation number one apply de Morgan's law in equation number one so this will become negation P or negation Q now is it the required PCNF is it the required PCNF or is it the required PDNF see PDNF means what it is a distinction of mean terms okay it is what distinction of negation P or negation Q so definitely it is a not a PDNF okay so another question will arise whether it is a PCNF yes of course it is a PCNF because it is a max term is there okay max term is there so it is a PCNF having only it is a product of only one max term okay this is the correct answer for the example number two let us focus on the third example let us focus on the third example okay the third example here write equivalent forms of the following formulas in which negations are applied to the variables only obtain PCNF given formula same question is there and the problem statement is formula is different okay formula is now here the formula is different negation to bracket P conditional Q okay as we know that to obtain normal forms we need to remove conditional connect to by substituting an equivalent formula by applying de Morgan's law by applying de Morgan's law okay let us assume that this is equation number one okay apply de Morgan's law to P conditional Q so that will become negation P or Q okay so the formula will be equation number one will become negation into bracket negation P or Q say this is equation number two okay now next this is equation number two okay so if you remove this negation negation that will become P and Q okay now is it the required PCNF okay in a PCNF what we do we take conjunction of max terms it is the conjunction of variables so we need to add we need to add the or we need to convert this variables P or Q okay that into max term okay that can be converted okay that can be converted okay by using the formula P or F is equal to P and Q or F is equal to F okay here we are going to substitute F is equal to F is equal to Q or Q and negation Q and in the second for the second variable Q okay for the second variable Q okay will substitute P and negation P in equation number three so P or F and Q or F okay this is equation number three in place of F in the for the first term will substitute Q or Q and negation Q okay so after substituting that okay this will become P or P or into bracket Q and negation Q and Q or P and negation Q so this is equation number four now apply distributive law over here okay apply distributive law to the equation number four that will become this term simplification of the first term will be P two brackets will be there two terms will be there okay that is P or Q and P or negation Q P or negation Q and the second one is Q or P and Q or negation P now we have to use committed to law okay and I don't put it law over here committed to law so after using that committed to law and I don't put it law what will happen P or Q is repeated twice okay so we know that P or P is P and P is equal to P okay with that formula so the one P or Q will be retained okay so finally only three terms will be there so this is the required so this is the required PCNF okay this is the required PCNF for the problem number two okay now now in this equation number six in this equation number six okay we have used I don't put it law okay we have used committed to law what is committed to law it was Q or negation P okay it was Q or negation P that has converted into negation P or Q and P or Q okay P or Q that was converted or Q or P was converted into P or Q okay that is that was a committed to law and I don't put it law is what P or Q okay the first term and the third term are identical okay so we know that P and P is equal to P so that's why This equation number 6 will become only three terms where P or Q and P or negation Q and negation P or Q. So this is required PCNF for the given problem statement. So these are the references and I hope you have understood now and you will be able to solve or you will be able to convert the given formulas into PCNF into principle conjunctive normal form. Thank you. Thank you very much.