 In the last class we discussed one of the important decision procedure methods that is natural reduction method earlier we also discussed truth table method and the semantic tab looks method. So in this lecture we will be talking about another different kind of decision procedure method which is called as reducing a given preposition logical formula into its conjunctive and decision to normal forms. So either you can reduce the formula into disjunctions of conjunctions that is DNF or conjunctions of disjunction CNF. So what we will achieve with this particular kind of reduction of the formulas into disjunction and conjunction of normal forms it is simple that so whenever you come across a conjunctive normal formula suppose X and something else P or not P etc and all we will be talking about two different kinds of formulas and all. So we will be reducing the given preposition logical formulas into either conjunctions of disjunction or disjunctions of conjunctions and all. So in the conjunctive normal form it is like C1 C2 C3 etc and all whereas each conjunct is nothing but a disjunction D1 D2 etc and all D3 D4 etc. So whereas in the case of DNF what we have is it is disjunctions of conjunctions and all. So we have conjunctions of disjunctions and disjunctions of conjunctions and all. So we go into the details of it in a while from now. So now observe this first kind of formula so in this formula the basic idea of reducing the formulas into conjunctive and disjunctive normal forms is this. So we know that in the case of conjunctive normal forms suppose in the case of this thing if a conjunction C1 and C2 C3 etc. So this is going to be true only when each conjunct is true and all otherwise it is going to be false. So that is the semantics of P and Q. So this reduction of well-formed formulas into conjunctive and disjunctive normal forms also follows this particular kind of idea. So that is TTF and F TFTF and then this formula is going to be false only in this case in all other cases it becomes T. So that means if you have conjunctions of disjunctions and all then if you come across in each conjunction you come across a literal and its negation and all then obviously the formula is going to be true. So this formula is going to be true this is true this is true every formula is going to be true provided when you have a formula and its negation is already there. So in the same way in the case of disjunctions in the case of DNF in particular whenever you have a formula a literal and its negation is there then it makes the whole formula false that means it makes the whole formula unsatisfiable. So this is the main idea that we will be using it so what essentially we will be doing in reducing the given well-formed formula into conjunctive and disjunctive normal forms and talking about the satisfiability and of course validity what we will be essentially doing is so we will try to reduce the given formulas which occur in this particular kind of format. So all the implications etc for example if you have a formula P ? Q ? R so what you will do is you will remove this implication by using the definition so that is a ? B is nothing but not here either the reduced formula will have only disjunctions and negations or the reduced formula from the given propositional logical formula which originally includes implication etc not so that we will reduce it to only conjunction and its negations and all. So that is what we will be doing it in CNF and DNF and in the process we will be able to talk about whether or not the formula is satisfiable etc so the basic idea of reducing the given formula into conjunctive and disjunctive normal forms is this suppose if X is any well-formed formula in the propositional logic which is of this particular kind of format X R P R ? P we know that P R ? P is obviously tautology and anything which is a disjunction of tautology have to be true only so because P R ? P is already true irrespective of whether X is true or X is false that is going to be true only so that means the given formula is obviously valid and all under whatever interpretation that you give it for X either it is T or F the variables that exist in X so where even though it X becomes true or X becomes false since P R ? P is already true so that it makes the whole formula true. So now the other idea is that disjunction is also considered to be associative because purely disjunctive well-formed formulas is valid if it contains a variable and it is negation and it is a unnegated formula like P R ? P is there and that is obviously tautology you know that makes the other formula also true so for example in this case ? P R Q it should be and in particular should not be R ? P R Q and P R ? R so this is considered to be valid because negation and unnegated forms are there here so that makes the whole formula true the first formula true and the second formula also is true so that makes the whole formula true so this is nothing but P ? Q so this is considered to be valid since it can be grouped to give simply Q R ? R if you simplify this formula you will get Q R ? R so the existence of propositional existence of propositional variable both negative and unnegated is a sufficient condition for validity so it is also considered to be necessary condition for validity you know so what you mean by validity any formula which is considered to be true a tautology is considered to be a valid formula you know so now observe this particular kind of thing and all for example if you have a formula like this one X 1 X 2 or not X 2 and another formula X 3 or X 4 or not X 4 and X 5 or X 6 or not X 6 something like that so now observe this particular kind of thing is a conjunction of all these formulas and all so it is conjunctions of disjunctions and all so that is why it is C and F so now each one each conjunct consist of set of disjunctions and all so now in this case this is already true X 2 or X not X 2 is true so now irrespective of what whatever is there here this makes the whole formula true so now in the same way a little and its negation is already existing here so that makes this thing true so that makes the whole disjunct true that means this is true this is true and even this is also it is also the case that X 6 and not X this is also true so all the conjuncts are true so that is a case here so all the conjuncts are true that is why the result in conjunct is also true C 1 and C 2 and C 3 is going to be true that means a given formula is a tautology so once you can show that it is a tautology you know obviously can that is as good as saying that a given formula is a valid formula so that is the reason why we reduce a given well form formula which is in the complex form which includes implication double implication etc into the normal forms which includes only conjunction and negation as its literals or disjunction or negation as the only things which you see in those normal forms so now existence of a propositional variable both negated and unnegated is a necessary and sufficient condition for inconsistency of pure conjunctive normal forms for example if you come across P and not P in the disjunctions in particular so this formula is going to be like this so in the case of this formulas for example if you come across in the disjunct in particular you come across a literal and its negation and all so that makes the whole conditional falls in a sense that P and not P is false so each disjunct will become false and a disjunction is going to be false only when both disjuncts are false in all other case it becomes T so in that way we will be able to know that when a given formula is also unsatisfiable and all so provided in a literal and its negation exist in that formula so a conjunct is true when it is all its conjuncts are true that is what is the case which is explained on the board on the left hand side whereas a disjunction is false when both the disjuncts are false so now what is a CNF so CNF stands for conjunctive normal forms that means a given well form formula you reduce it to conjunctions of disjunctions so now it is D1 and D2 and D3 where D1 D2 are disjunctions and all so disjunctions are represented by R and conjunctions are represented by and so if it is conjunct of a B then it is represented as A and B disjunct means A or B so now a well form formula in a preposition logic D1 and D2 and D3 and DN is said to be in conjunctive normal forms especially when X is considered to be an unnegated conjunction that means if it is a unnegated conjunction means it is P and Q etc and all if it is a negated conjunction it will lead to disjunction and all so that is why you have to rule out this possibility that X has to be an unnegated conjunction and every conjunct in the disjunction D1 D2 D3 etc. and all in X is an unnegated disjunction and all and the third condition is that every disjunct is considered to be a propositional variable it has to be either a propositional variable or it might occur in the form of negation of a propositional variable it should be in the form of P or it should not be in the form of not P. So in short X is a conjunctive normal form even only if it is a conjunction of disjunctions and no negation sign has an argument other than the singular propositional variable. So the idea here is this particular kind of thing first we are trying to talk about conjunctive normal forms so what is the conjunctive normal form it is the conjunctions of disjunctions D2 D3 etc Dn so D1 and D2 D3 Dn etc are considered to be conjunctions of disjunctions. So now the idea here is that this conjunctive normal form should have these things that formula X is an unnegated kind of conjunction for example if you have something like not D2 or something like that for example let us say D1 is P1 P2 and let us say P3 or P4 etc. Now let us assume that these are the only two formulas that are there and so in the pure sense this is not considered to be a CNF conjunctions of disjunctions not because it consists of negation of some kind of formula and all negation of disjunction and all so that means this formula has to be reduced it will be reduced to not P3 and not P4. So if it consists of negation of some kind of variables then that is not considered to be a conjunctive normal form. So you have to ensure that all these three conditions are satisfied and all X is an unnegated conjunction and all and every conjunct in X is an unnegated disjunction that is the thing which needs to be followed and every disjunct in a proportional variable or the negation of it should have to it is either a proportional variable or it should be a negation of that particular kind of variable. So let us consider some examples and all before that a well formed formula is considered to be a degenerate conjunction it appears like we have used these three rules and all like X is an unnegated conjunction so that is satisfied in this particular kind of formula P or not P or Q actually this is a disjunction but it looks like it is also it can also be considered as a conjunction also I mean conjunct C1. So X is an unnegated conjunction there is no negation of a formula which is not of X and Y etc and all which occurs there in this formula P or not P or Q and every conjunct in X is an unnegated disjunction here we do not have any this negation of disjunction kind of thing occurs here and is not of P or P etc will not figure out here second rule is also followed and every disjunct is a propositional variable or the negation of a propositional variable either P is the case the first one you have P and the second case you have not P so literal and it is negation is there and all this so every disjunct is a propositional variable or the negation of a propositional variable and it satisfies this particular kind of conjunct which is written in the form of a disjunct satisfies all the things and all it is in that sense this is called as degenerated kind of conjunction so usually it is a disjunct in all but the whole term P or not P or Q can still be treated as if there is only one such kind of formula which exists there here it can be called as a degenerate case of a conjunction so why because we can still we can write like this of course this is P is a conjunct P or P or not P or Q maybe if you want to extend it you can write it as P and P or not P and not P which does not make any sense and all but still you can write it in that way Q and Q etc Q and Q is same as Q etc or you can even write it as PR in various ways you can write it and all but this is an example of degenerate case so what essentially we are trying to talk about is this particular kind of things CNF so what is a CNF first of all it is a conjunction of disjunctions and all this is conjunctions of disjunctions and all so D1 D2 D3 etc so now each one is called as the degenerated case of this one is PR not P or Q because it follows all the three rules of the one which we have talked about so now some kind of convention which we follow for coming up with CNF and DNF of a given well formed formula so the conventions are like this no redundant disjuncts occur in any conjunct in a CNF conjunctive normal form so that means if you come across a formula such as PR P or PR P something like that it is simply represented as P however number of times you it occurs in all in the formula three times four times it does not matter but it is treated as P only because you are not saying anything more than P here so redundant disjuncts will vanish in all so now we will just write it as PR P so it is like in the Boolean logic we have one R1 that is 1 plus 1 is 1 only so the same thing which is followed here also so the only thing which we are talking about here is that instead of disjunction we are talk instead of plus operation we are talking about disjunction so that is the first thing the convention which we follow and the second thing is no redundant conjuncts occur in the conjunct in normal forms just like P and not Q and suppose if you have another P and not Q it is reduced to P and not Q even if it occurs 100 times in all but it is still treated as P and not Q the third thing is that within disjunctions that is D1 D2 D3 etc and all variables appear in the alphabetical order if it does not appear in the alphabetical order you need to use distribution laws or associative law and make it in some particular kind of order so now particular order with the only unnegated occurrence of a variable preceding the negated occurrence of a variable it is like so that means unnegated ones should come first whereas the negated ones come later so now for example observe this particular kind of thing not PR R or not Q or Q or P and etc and all so here negative forms came first and then unnegated form came later etc and all we need to reorder this particular kind of thing suppose if you follow this particular kind of convention like ABC etc and some order we will follow so now this will change to so the first disjunct it is not PR R or not Q or Q or P changes to PR Q or not Q or not Q or R and so we are reshuffling this particular kind of thing and then we are putting it in some kind of order after P Q follows Q or follows only for our convenience you know so these are the three thing conventions which we follow so that you know we will avoid a lot of redundancy etc so now how to convert a given well form formula into its corresponding conjunctive normal form in the same way you can apply the same procedure to reduce the given well form formula into disjunctive normal forms where a disjunctive normal form is disjunctions of conjunctions so first what you need to do is we need to use some definitions to remove the occurrences of implication double implication and R so that you know it will be converted into conjunctions you know so then you use De Morgan's laws and the law of double negation that is not not P is equivalent to P to remove all the negations which are there outside the brackets and all for example if we have not of X and Y it will become not X or not Y the same way not of X or Y will become not X and not so now then you have to use the double negation to ensure that no variables preceded by more than one negation sign that we need to ensure and then of course we need to follow some other kinds of rules such as distributive law PRQ or R is nothing but PRQ and PRQ or PRR actually this should be PRQ and R or you can use P and Q or R if it is PRQ and R it will become PRQ and PRR if it is P and Q or R it will become P and Q or PR P and R P and Q or P and R so these are the distribution laws which we apply to this one so that in given well form formula is converted into appropriate conjunctive normal form. So now let us try to convert this particular kind of formula into the conjunctive normal form so P ? Q ? P is what is given so now first thing we need to do is to eliminate that implication how do we eliminate this implication by using a simple definition of material implication that is P ? Q is nothing but not PRQ so that is a reason why the first conditional P ? Q ? P will become not PR again Q ? P is written as not Q or P that is the definition that we have used in the first step. So now this further simplifies to you can use associative law and then you can regroup or reorder it in such a way that it will become PR not PR not Q so this is kind of case which we have talked about earlier that is a degenerated case of the special case of conjunctive normal form there is a degenerate case where you have all the rules three rules are applied on this particular kind of thing. So now this PR not PR not Q is finally considered to be in that particular kind of conjunctive normal form so this as such in it appears to be not to be in the conjunctive normal form but you can write it in this way that PR not PR not Q and the same thing which you can write it PR not PR not Q so now this is nothing but this is like C1 and C2 where each C1 is nothing but a degenerate. So in this case PR not PR not Q because PR not P is already true and all this because we know that it is a tautology so anything whether Q is false or Q is true that is going to be true only so it is in that sense it is considered to be a valid formula also of course this is indeed a valid kind of formula in the preposition logic because it is an instance of paradox of material implication in the last class we showed that using natural deduction we showed that P ? Q ? P is a theorem or it is considered to be a valid formula. So what essentially we did here is that given a formula we reduce this we eliminated this implication by using De Morgan's laws and the definition of material implication etc. and all and we reduce the formula into a formula which consists of only disjunction or negation so these are the only things which you will come across in the final formula. So some more examples which we take into consideration how to reduce a given formula into conjunctive normal form so conjunctive normal form is conjunctions of disjunction each conjunct these combination of different formulas which are in the form of disjunction. So what is given to us is P ? Q ? Q is what is given to us so now what essentially one needs to do is in this formula we have implication and conjunction is there so now we need to eliminate this implication how do you eliminate this implication use the definition so that is a ? B is nothing but not here so this is P and P ? Q that is taken as X I mean that means not of X or Y Y is same as Q here so that is what is the case in the what is happening in the second case so now once you use the definition so now if there is any negation outside the bracket and all you need to push it inside by using demagans laws so now in the second step the negation is outside the bracket and now we push it inside negation of conjunction will become disjunction negation of a formula P you will become not P ? P ? ? ? P R Q that is the first one R Q is the case this is not still in conjunctive normal form that means not conjunctions of disjunction but you have to reduce it further using further applying demagans rules and whenever you have ? ? P X I now use the notion of double negation so the first formula ? P R ? ? P R Q will become now P ? Q and Q remains the same so now it is not not P now use distributive law X R Y and Z will become X R Y and X R Z that is what we have done here not PR P and not PR not Q or Q and Q is remains the same so now once you have done this thing what you need to do is you have to reorder the formulas in such a way that there is some order which is maintained the convention we need to follow P preceded by Q etc so now this Q ? PR P and ? PR ? Q and the Q if you use distributive law it becomes Q R ? PR P and then Q R ? PR ? Q so now you drop the brackets and all so now this again you can use some kind of reordering it will become you can use associative law here it will become PR ? P and Q so you have to ensure that unnegative term comes first followed by that negation of its term and then followed by that anything goes in the same way in this case also ? PR Q R ? Q that means first unnegative term came first and then the negated term comes later so this is what we have reduced a given formula P and P ? Q ? Q is reduced to this one so what is so great about reducing this particular kind of formulas so now observe this particular kind of formula this is like a X and Y kind of thing so X and Y when it is going to be true when both conjuncts are true so now in this case if you observe the first conjunct that is that involves PR ? PR Q PR ? P is obviously total so anything whether Q is false or Q is true that is going to be true only that means the first conjunct is automatically first disjunct is automatically true so now coming back to the second one here also we have a literal Q and its negation is there so that means is always true now irrespective of whether not P which occurs there in this formula second formula whether it is true or false that is going to be true only so that means the entire conjunct C1 and C D1 and D2 both are true so that is why the given formula is true that means a given formula is a tautology so how this happened and all especially because each disjunct has a literal and its negation occurs in this particular kind of disjunct in a conjunctive normal form so that makes this particular kind of formula true that means tautology and hence the given formula is a valid kind of formula so now a well form formula is in CNF especially when it has this form it is a conjunctions of disjunctions first you need to write talk about the connective that occurs there so that is a conjunction of what conjunction of disjunctions that is why D1 and D2 to DN where n is greater than 1 whereas a is considered to be a well form formula which is in disjunctive normal form the other way around the dual of this one if it is a disjunctions of conjunctions so so how you need to read this particular kind of formula sometimes it will be confusing for us in the case of conjunctive normal forms so what appears here is it is a conjunctions of disjunctions so now DNF is disjunctions of conjunctions C3 say conjunctive normal form is going to be true when all the disjuncts in that occurs in that conjunctive normal form is going to be true so when this happens especially when you have a formula like a literal and its negation occurs in this particular kind of formula then each disjunct will become true then that makes the conjunctive normal form true and hence valid in the same way DNF in the case of DNF each conjunct has to be suppose if each conjunct is false then that makes the whole formula unsatisfiable enough so for showing the validity you will use CNF for showing the unsatisfability you will be using the DNF so DNF disjunctive normal forms are nothing but disjunctions of conjunctions first you need to read the sign here this is a disjunction of what the conjunctions in the same in the conjunctive normal form first you write the conjunction here conjunctions of what disjunctions D1 D2 D3 etc so now there are some remarks which needs to be discussed here so a well form formula is in disjunctive normal form if it is an unnegated conjunction of variables that means there is no formula which occurs like this that it is not of P or Q etc and all whereas individual letters will have negation and you can have P or you can have not P also so either negated are unnegated the procedure for finding DNF is exactly same as that of CNF that means you remove the implication sign first by using the definition and then you push the negation inside and use De Morgan's laws and distributive laws etc then ultimately you will have in the case of DNF's you will have disjunctions of conjunctions in all and if every conjunction in DNF contains some variables and negated and unnegated then the given well form formula is said to be inconsistent so that means in any formula like this thing they are all conjunctions and all so like X1 X2 X3 etc so they are all conjunctions so now it so happened that you have X1 and not X2 in any one of this in all this conjunction also so that means this this makes this formula this makes this C2 falls in the same way C1 is also having that particular kind of thing a conjunction and a literal and its negation occurs in each conjunct in all that makes every conjunct falls and since since all the PRQ is going to be falls when both are falls that means every conjunct is falls then this is going to be inconsistent or unsatisfiable kind of formula. So now the question comes to us is there may be some kind of perfect normal forms a CNF a conjunctive normal form is said to be a perfect normal form especially if every conjunct contains as a disjunct every propositional variable negated and unnegated that occurs in the whole well form formula for example if you take example will serve a purpose here so PRR and PR0Q so this is an imperfect kind of normal form so what one needs to do here is that we need to find a perfect normal form corresponding to this imperfect normal form so how do we find it there is a procedure for finding it so if in any conjunctive normal form D is a conjunct that means D is a conjunct which does not contain some prepositional variables pK then D is represented as DR pK and DR pK in if in any disjunctive normal form C is a disjunct which does not contain some prepositional variable pK so how do we reduce an imperfect normal form into a perfect normal form so the first one is this thing PRR and PR0Q for example let us take into consideration this thing so now this formula PRQ and NOTQ is written in this sense PRQ or PR0Q so now this is p is same as dropping the inconsistent disjuncts and all what is the inconsistent disjunct Q and NOTQ if you drop that particular kind of the Q and NOTQ is obviously false and all and PR false leads to P only so now P leads to P is nothing but PRQ and PR0Q dropping the inconsistent disjuncts so now PRQ and PR0Q that is the first one and then the second one PRQ and PR0Q are all so this is another kind of thing in perfect form that is translated into this PRQ or all using the distributive law and you can write it in this way PR0Q or all so now again this reduces to P PRQ or PR0Q or all so now P is equivalent to PRR and R so it is R and R is reduced to F so PRF is P only now PRR and PR0R so now PR0Q or R and PR0Q or NOTR so now we have after doing all these things we have something called a perfect normal form so what is that we have done here is that we have reduced the given formula into corresponding kind of perfect forms in perfect forms have been translated into perfect kind of forms so something is in imperfect conditional in perfect well-formed formula can be reduced to perfect kind of formula PRR and PR0 so now so this can be written as this one so the first thing P it can be written as this one using disjunctive normal form this is nothing but PRQ now this we are trying to translate it into perfect kind of form and all so now PR you are adding this particular kind of thing Q and NOTQ this is nothing but PRQ and PR0Q so this is using distributive property PRRQ and PR0Q and all this is first is PRQ and PR0Q so now this reduces to just P so now what we mean by P is this one PRQ and PRQ so what exactly we are doing here is is that since it is not in perfect normal form and all conjunctive normal form where this is D1 and D2 so we are trying to reduce it into perfect kind of normal form so now using this particular kind of equivalence we substitute it in this particular kind of thing so now this P is substituted here so now what will happen PRQ and PR0Q RR so this is the first disjunctive so now this will become this one this is PRQ RR and this is and PR0Q RR so this is the second disjunctive so how did we get this particular kind of thing first we included this particular kind of thing PRQ and NOTQ this is same as P only we are not just doing anything new way so this is nothing but PRQ and PR0Q so now the same thing whatever we generated here this we substituted in this one so this is what we have written here our distinction are so now this is what we get so now the same thing you substitute it into the second one you will get so you will generate the other one PR so now what you will do here is so now PRQ RR and PR0Q RR in the same way you can write P so now P as so this is same as P only so now this can be written as PRR so now you substitute this P value whatever we got it here into the second one so now this becomes for P you substitute this one PRR and PR0R this is the first thing the whole thing which we got it for P is nothing but this one the whole thing are not so now this is as it is in all and so now this will become PRR R0R and PRR PRR PR0R so PRR R0R this is the first thing and then you have to use this PR0R so now this seems to be in the perfect kind of normal form so initially we have just PRR and PR0Q so now you are if you are trying to write it in the normal form and all we need to ensure that all the disjuncts that occur here will have all these letters and all whereas in this case Q is missing here whereas in this case R is missing so this is not in a this is not in a perfect form and all a perfect form includes all the disjunction of whatever disjunct occurs in the given formula so for that reason what we have done is simply this that we have just taken one formula into consideration which is an equivalent of this particular kind of thing P for example if you have X R P and not P is same as X only because P and not P is obviously false and all X are something which is false is X only it is like X plus 0 is X so now essentially what we have done here is that this is an imperfect normal form in the sense that the disjunct D1 does not consist of Q the variable Q here in the same way in the case of disjunct 2 we do not have R here so we need to transform this thing in such a way that if you want to have it in a perfect normal form then we need to make use of this particular kinds of transformations and ultimately you will see here this is the final translated version of the same formula PRR and PR not Q that is like this PR Q or R and PR not Q or R and followed by that we have PRR not R and PR not R and not R but now here so can we say that this particular kind of formula is valid or not so for validity what needs to be the case is that all conjuncts have to be true so that means each disjunct have to be true to make this formula true so we can talk about such kind of thing for only these two formulas PR not Q or R at least a literal its negation is there not its negation some negated formula or negated formula is there but when it comes to this particular thing PR not PR Q or R if this any disjunct will become false it is going to make the whole conjunct falls had it been the case that if it has at least one not R here and then if you have some P and not P or something like that is here then it makes all the conjuncts all the disjuncts true and it makes the conjunctive normal formula true. So this is the way in which one can convert a an imperfect normal form that is PR R and PR not Q into this particular kind of format and all where we have not changed the entire structure of the formula and all but we made use of some definitions and all like P is nothing but PR Q and not Q and in addition to this particular kind of thing definition and pushing the implication inside etc and all so what we are trying to do is we are making use of some laws of logic also so one is the first one is law of identity P and something which is a tautology is obviously P only or identity law is not identity law but PR the contradiction is obviously P only it is like in the Boolean logic PR 0 P plus 0 is P only and the domination law if you add anything any contradiction to a given formula contradiction dominates and all so that is given statement is going to be contradiction and the domination law with respect to disjunction if you add any tautology to a disjunction disjunct that is going to be true only the tautology dominates over here in the first case contradiction dominates there an idempotent laws P and P is nothing but P PR P is also P only and commutative laws as we know P and Q is nothing but Q and P which happens which occurs for even disjunction also PR Q is nothing but Q or P on associative law P and Q and R or PR Q or R the same as PR now brackets Q or X so these are some things which we already know and the De Morgan's laws are as usual not of P and Q will become not P and not Q and absorption law is simply seems to be little bit surprising for us something which is PR P and Q will become P so now some more examples how to convert not of P implies Q or R implies P into its corresponding conjunctive disjunctive normal forms the first step what you will do is you write we remove the implication so then P ? Q will become not PR Q that is followed by negation of that particular kind of formula or R ? P is written as ? R P so now in the first disjunct first disjunct you have to push this negation inside then it will become P and ? Q and R ? R P remains the same so just let me work it on the board and all so that you will get to know this thing in a better way just one formula which I work on it the rest of the formulas one can transform it into the appropriate DNF so the formula that we are trying to reduce is this thing not of P ? Q implies not R P so now we are trying to reduce it into CN the CNF is like this dis conjunctions of disjunctions D3 etc so now the first thing which you do is you have to eliminate this implication and all so now how do we eliminate this thing so we have a definition P ? Q by definition is nothing but not P R Q so this is what you write it here and then you remove it in the next step you know so now this is not R P so now in the second step you remove this particular kind of implication so now this will become the whole thing is treated as X and this as Y X ? Y is nothing but not X or Y so now we substitute for X this is not of not of not P Q or not R P so now this is now not not of for this one will become this one not P R Q or not R P so now you can use distributive law and all then you will get not P R so not P R Q or not R R not P R Q R not of P ? Q and then R P not of not not of not P ? Q yes here thing is that not of not P R Q it will become and negation of a disjunction will become conjunction of this is as it is so now this needs to be changed like this not P and Q or not R R P so now this step will become this thing so now you apply this thing not P and Q P and Q and not R R not P and Q and not P and P that happening so now it is not P and Q or so now this will become not P R so now this this thing we take into consideration and you remain keep it like this only not R or P and Q and is the one which is coming here so now Q R not R P so now this is in the disjunctive form D1 and this is D2 now it is a conjunction of D1 and D2 that is why it is called as conjunctions of disjunction that is why CNF conjunctive normal form so now whether or not this formula is satisfiable or not this is not in not in a perfect form and all now we need to reorder it and then you can write it like this so not P R P now this comes first unnegated term comes first followed by that you have a negative term and then not R and in the same way Q R not R you write it in this way P R Q R not so now in this case you have P R not P which is already true is a tautology so now irrespective of whether not R is true or false is going to be T so now even in this case also P R Q R not R a negative term and the unnegated term is there here so that means it ensures us that whether or not P is false or P is true this whole formula is going to be T on so that means the whole formula is going to be true that means this formula is going to be valid form so in that sense we can talk about validity of a given formula especially in the form in the case of CNF so in this class what we have seen is that a given well form formula we reduce it into its corresponding conjunctive or disjunctive normal form conjunctive normal form is nothing but it is a conjunction of disjunctions and disjunctive normal form is disjunctions of conjunctions so the basic idea of reducing this conjunctive given formula into conjunctive and disjunctive normal forms is that the movement you see the structure of this formulas you can immediately come to know that using the semantics of disjunction semantics of conjunction you will come to know especially in the case of CNF if each D1 is true then the conjunctive normal form is going to be true in the case of DNF's it is a disjunctions of conjunctions if each conjunction is false then you can show that a given formula is unsatisfiable and so this is the advantage of reducing the given formulas into given well form formulas into its corresponding disjunctive or conjunctive normal forms and sometimes it might appear that these formulas will not appear in a perfect form like all the variables will not occur in a given formula so we need to do a little bit of transformation without disturbing the truth value of it just like you know for example you can replace P with P or Q and Q Q and not Q so that is P or Q and P and not Q and all. So like this you know one can transform in perfect normal forms into perfect normal forms one example we have seen in this particular kind of class in the next class what you are going to see is that we will be making use of this conjunctive and disjunctive normal forms especially in solving some kind of puzzles as well as will be making use of this conjunctive normal forms and disjunctive normal forms in analyzing some simple switching digital circuits. So we will talk about analysis of as an application of this conjunctive and disjunctive normal forms we will be talking about the analysis of simple digital switching circuits.