[Music] we have read the predicate logic and the two quantifiers the universal quantifiers and the existential quantifiers and mainly how we can write find the truth values of the statements using this 12 what if s now we read the generalized predicate logic and the de Morgan's law so today we see that generalized predicate logic first we see that how the same statement can be represented using the existential quantifier as well as they using that universal quantifier or whether at all it is possible so first we see one simple example we take one simple example like say there exists X X by X square plus 1 is greater than 1 say this is one existentially quantified statement and X can take the value from the set D that means D is the domain of discourse which you can take some positive integers so they are the positive integers now since it is a existentially quantified statement so if we get one such X for which is X by X square plus 1 greater than 1 then the statement is true now for any positive integers since X takes any value X takes value from D so for all values of X in D this X by X square plus 1 we know that never is greater than 1 because x squared plus 1 as X square plus 1 is greater than X for positive integers for all X so for all values of X this is false so these existential quantified statement is false this is false no I can write if I what we have seen that for all values of X this is not true this is false so then can i if I if I just invert this thing that or we can take negation X by X square plus 1 greater than 1 so what is that that is equivalent to that X by X square plus 1 less than equal to 1 so now now can we check this thing that for what happened for all values of X these X by X square plus 1 less than equal to 1 now since it is the universal quantifier for all X so if it is if we have to show that it is true then for all values of X in D this should be true now see that this is the negation value this is the value of negation and we see that for any value of x or or for X any value of x indeed this is true X by X square plus 1 is less than equal to 1 actually it it is less than 1 only if we consider the teks equal to 0 then it is then also it is it is less than one so if I take this is just change that if I take this thing and so greater than equal to one and negation we take negation we take then we take this is less than greater than so we take this is less than I think then we can tell that it is true that means for x value for x in d this is true now see that this is nothing but the negation of the statement and these there exist X what I have are real you have given there exist xpx and here this is for we have taken for all X negation px No we can write then we can write the same px the propositional function using the exist tential quantifier or using the universal quantifier but obviously that it will be the totally reverse or the negation here it will be the statement it will tell either it is true or it will be false so what we can tell from these simple example we can think that there exists X and or I should write there exist xpx and for all xpx the relation what they are connected or the relation between these two relation between these there exist X and for all X one negation is involved since in the last example what we have seen that X by X is squared plus 1 greater than equal to 1 and the negation is that X by X is squared plus 1 less than 1 and then we can write if 1 is written using there exists X then another we can write a 4 using for all X now how to find negation already we have read to find negation we use the de Morgan's law we use de Morgan's law that we have used in our propositional logic no here we see how we can use de Morgan's law for all the predicate logic so the Morgan's law for predicate logic since we have two quantifiers so here we must have two pairs of statements that should be equivalent so we can write that one is for universal quantifier so how to find that negation for all X px and this is their existence negation px second one is for negation there exist X px and for all X negation px so though for both these pairs they are actually equivalent so we have to prove this thing so this is the relation between the existential quantifier and the odd existentially quantified statement and the universally quantified statement and there they are related by a negation what I mentioned earlier that one negation is involved now we prove one of the statement or one of the de Morgan's law the negation for all X px whether they're equivalent there exists X negation px the first pair we see so we start with the left hand side so this is one universally negation of some universally quantified statement so our LHS is some negation of universally quantified statement so the statement can be true or false so first we consider the case that let the statement for all xpx is true all right if the negation check negation for all xpx is true so for all X px if we omit negation so the reverse is this should be false now according to the definition basic definition the basic definition of the universal quantifier for all X the universally quantified statement is false if for at least one value of x in D px is false so since it is false that means there exists at least one value of X for which PX is false so how by using the notation existential quantifier this statement we can write there exists at least one that means there exists X px is false that means negation px is true px is false which tells that or we can tell that there exists X negation px is true so negation px there exists X negation px is true and we started which is nothing but our right hand side so what we have considered that if the negation for all xpx is true and we see that there exists X negation px is true which is our these de Morgan's law for all now another part is that we have to consider if it is false now we see that if we see the B part that if negation for all X px is false so if this is false so obviously similar way we can tell that for all xpx is true now for all extra px true that means that for all values of X in D since it is a universally quantified statement so for all values of X in D P X is true now for existentially quantified statement if it is a true then what will then for all values of X this can be false that means that for all values of X px is true so for the all values of X negation px is false which is nothing but our our richest and this is the left hand side we started for the other case when it is false so I can I can just write this is this is our true for when we have considered that it is the universally quantified statement is false this is false and here we can tell that whenever the our universally quantified statement is true so both the cases we can prove and this is our demand answer so similarly we can similarly we can prove the existence she want effect statement that is there exist X on negation of there exist X px is equivalent to for all X negation px similar way we can do so we see the de Morgan's law de Morgan's law we can apply to find the negation of the universal quantifier statement or the existing Shelly quantifier statement now how we can generalize this thing you can write the generalized statements so the concepts the basic concepts that we have applied to prove the de Morgan's law that we see the generalized de Morgan's law on generalize predicate logic so let's we'll consider a universal quantifier statement for all X px and X can take value from the domain of discourse D that means Witcher D is the value of D 1 D 2 up to DN and let the statements of the propositions our P 1 P 2 P 3 P n which is nothing but the P of D 1 when X takes the value D 1 D 2 like P D n that means when X takes value D 1 from D 3 D 1 which is P 1 which is which can be either true or false that is a proposition and similarly for p.m. now if we take the conjunctions of all such people all propositions that means we take the conjunction so P 1 and P 2 up to P so what are the physical meaning of this conjunction this physical meaning of this conjunction this is equivalent to for all X P X because for all X means for X Tech's each value from D that is D 1 to D 10 the N and for death the statements become so the proposition becomes P 1 to P n so if I take the conjunction of P 1 P 2 P n then this is for all X P X so similarly if I take the disjunction if I take the disjunction so this is my the conjunction if I take the disjunction so this will be P 1 or P 2 or P n and you know we can write that this will be there exists xpx when that any value one value of x that means one of P is are true so we can now we see what we can tell from here we give as if these are the conjunctions of all the propositions are the disjunctions of all the propositions or little firstly we see how they're whether they're true or false so we immediately we do not write the universal or existence yet statements we see that now from the definition of for all X what do you know that good for all xpx we write that for all X px is true if for all values of x in d PX is true so for all values of x in d for all values of x the propositions become so i can write then for all values of x PX that i can write that is equivalent to p1 conjunction p2 conjunction p3 conjunction p.m. so when it will be true since it is a conjunction so if all are true then only this is true that means LHS will be true if this conjunction is true this part is true if everyone is true now all PA values of all P eyes are true all P Iser you can write all P is a true then only I can tell that this is are true and since it is conjunction so it will be false it will be false when any one or at least at least one P I is false so this is my generalized form of predicate logic which is nothing but so what we can tell that what we can conclude that for all X P X is nothing but the the conjunction of conjunction of all the of all propositions now if we see the there exists X so definition of there exists X px when it will be true there exists X px is true if for one value at least one value of x in D px is true and false when for all values of X px is four and four so I can write is p1 or p2 or PN since we know that it will be true for this will be it is true if one value of at least one value of x in D px is P I is true that means px is true that means one p is true and when it will be false when it will be false if for all X in D px is false that means all pies are false that means this P or not P 2 or P n this con disjunction is this disjunction is possible so we can generalize our existence really quantified statement or the universally quantified statement by only using our conjunction connectives or the disjunction connectives of the propositions or the platelets so any once we know these generalized de Morgan's law so any number of predicates that we can relate or the with the existence real quantified statements or the universally quantified statements any number of quantified statements that we can relate with the simple the negation rule and the number of conjunctions and the number of disjunctions of this thing