 Hello, welcome back. I am Professor R. V. R. Giddi from Computer Science and Engineering Department of Valachand Institute of Technology, Solapur. Today I will be focusing on the various examples on principal consent to normal form that is PCNF. In the previous videos we have seen the examples on PCNF. The learning outcome of today's video at the end of this lecture student will be able to convert the given equation into principal consent to normal form. The examples of various, the possible mean terms, the possible mean terms for two variables they are PORQ negation PORQ POR negation Q. This is just revision okay these are the, this is set N set B from this set two sets we have to consider one of the set as a maximum and we have to take in a product of those maximums okay and these are the possible maximums for the three variables PQR that we have already learnt in a previous videos. Now today take this question okay how to order the possible maximums for three variables and variables are say PQR. Please take a pause over here and write down the order of those maximums. The order of the maximums will be start with capital M as a subscript zero and that subscript zero is converted into binary terms, binary bits zero zero okay and this is how the variables are written that is POR this zero zero is written because wherever zero is there okay we have to write a variable in a product in a summation okay that the first bit is for P second bit is for Q and third bit is for R okay this is how we have written all the maximums and we have ordered all the maximums okay and the maximums in the last maximums okay that is M7 is nothing but M subscript is 111 indicating that all variables are present all variables are present that is PQR with a negation P or negation PQR negation R okay when you take when you take this in a product okay when you take in a product that will be PCNF for the given formula. Now let us see what is the problem statement okay problem statement is we have been given S is equal to negation P conditional R and Q by conditional P okay and we are supposed to obtain PCNF okay now what we will do first of all to convert the given formula to convert the given formula into consecutive normal form or any normal form first of all we need to remove the higher connectives like conditional and by conditional by substituting and equivalent formulas and that formula and then we have to convert that formula into formula such a way that it will contain only basic connectives like and or and negation. So let us do it that okay let us assume that S is equal to say equation number one okay for negation P you can use de Morgan's law over here that is P conditional R is equal to negation P conditional R is equal to P or R okay since we know that P conditional Q is equal to negation P or Q okay using that particular formula P conditional R is can be converted into P or R okay then second formula Q by conditional P okay that can be converted or the equivalent substitute formula is Q conditional P and P conditional Q okay let us substitute this substitutions in the equation number one. So this will become P or R okay this first term in the equation number one will become P or R and Q conditional by conditional P will become Q conditional P and P conditional Q in the next slide okay in the next slide we will see how to remove this conditional from Q conditional P and P conditional Q so that can be done by P or R will be remain as it is so Q conditional P will be converted into negation Q or P and this will be converted into negation P or Q. So in the equation number three the terms whatever we observe they are not a max terms because max term is what it is a summation of all the variables which are present in the given formula if you look at the given formula the given formula contains three variables P, Q and R okay P, Q and R so it contains every term they are in a product that is okay but the term P or R is not called as a max term because it is missing the variable Q and negation Q or P okay it is missing the variable R and the third term negation P or Q it is also missing the variable R. Let us add that particular variable with respect to that particular max term okay for that what we can do we can use the formula some equivalent formula P or F is equal to P and for F we are going to substitute over here F is equal to okay F is equal to we are going to substitute Q or Q and negation Q here we are going to substitute Q and negation Q okay now let us see okay F is equal to in the first term for the first term okay it is missing Q so F is equal to put Q and negation Q for the second variable or the second term negation Q or P this is missing the variable R. So let us add F is equal to R and negation R and for third term okay it is missing R so again we put F is equal to R and negation R in the equation number three okay after substituting F is equal to the respective terms okay equation three will become now P or R okay R, Q and negation Q then the second term R and negation R and third term R and negation R this will become equation number four okay now we have to apply distributive law between these two terms and then simplify the formula we have to use distributive law, idempotent law okay let us use distributive law and idempotent law equation number four okay so that equation number four now will become P or Q or R okay P or Q I have taken P or R or Q okay so while putting also I am using distributive law as well as associative law that is P or Q or R okay in one bracket then and P or R or negation Q that can be written as P or negation Q or R okay after applying distributive and associative law this the first term is simplification of the first term is into two terms then the simplification of these two second term will be P or negation Q or negation R okay so this order is changed over here okay after applying distributive law and associative law okay these two terms are there then finally negation P or negation Q okay and we have removed one particular term okay 1 2 3 4 5 terms are left okay so this is a required PCNF this is the required PCNF after applying distributive associative and idempotent law in equation number four now apart from that PCNF apart from that PCNF okay also we will find out PDNF from this given state of the formula okay suppose this is equal to say S this is equal to say formula S okay so what we will do now the conjunctive normal form of negation S can be obtained can be obtained by the conjunction of the remaining max terms thus negation has the principle conjunctive normal form so negation S okay so negation S means what we have done here those ordering of the max terms okay the red color indicates that the red color indicates that all the max terms are utilized in a PCNF okay now I have to take the black color that is P or Q that is m1 okay then I have to take m6 and m7 okay product of this remaining max terms okay that is equal to negation S okay that is m1 m6 and m7 okay that is equal to negation of S okay let us write down that particular negation S is equal to this is equal to negation S that is the product of remaining max terms m1 m6 and m7 this is equation number six now to obtain PDNF to obtain PDNF okay to obtain PDNF what we have to do we have to take the negation of negation S negation of negation S negation of negation S means we have to apply negation to entire this particular formula okay to the equation number six to the equation number six so once you use that negation to this first term, second term and third term okay after adding this negation by using de Morgan's law okay when you add this I will explain the first term that if you take negation insert the bracket this p will become negation p okay and r will become and this q will become negation q and this r will become negation r this is how the remaining max terms are solved and we have obtained the equation number seven that is nothing but the required PDNF required PDNF so in this example what we have seen we have taken a formula problem statement S okay from that we have obtained PCNF okay then from that PCNF that S is equal to we have taken negation S is equal to remaining product of remaining max terms and then negation double negation S that means that negation of remaining max terms okay that negation is taken inside the bracket so we obtain the PDNF so these are the references I hope you understood how to solve the problems on PCNF and or rather how to obtain PCNF thank you