[Music] let us continue in discussing predicate and quantifiers this video lecture will focus on binding variables and negating predicate and quantifiers so for example let p at x y be x less than y and consider universal x p at x y can we determine if the statements true or false and again the domain of x is number okay real number so can we evaluate this statement is this a proposition definitely not y we know the value of x we know the domain of x and this is true for all but how about for y do
we know the value do we know the group where it belongs to do we know the domain definitely no so therefore we cannot determine if this is this is true or false therefore it's not a preposition so we need to define y first before we could evaluate it okay so for example supply a value which is equivalent to five so evaluate this reversal x be at xy5 we substitute 5 to our equation does it satisfy our equation yes to a certain degree but not all values of x since not value not for all values of
x therefore the statement is false okay or instead of supplying a specific value let's create an equation x plus y for y definitely x plus 1 is greater than x so therefore any value of x y is greater than x because of this equation therefore the statement will be true okay so again if we have multiple variables it should be bound by a quantifier or we have a specific value to make it a proposition okay another example let's have this extension xp at x or 2 q at x is this a preposition definitely no why
because extension x exclusive for p at x therefore q attack q at x has no quantifier bounding it therefore we cannot determine if it's true or false it is not a proposition we need to have a quantifier here or a specific value for x okay for qrtx to make this statement as a proposition okay another example extension x be at x or universal x cubed x so this is the correct way of binding our variables it could be and we could we have an exclusive or b at x we have an exclusive for q at
x this is a preposition okay another example extension x be at x and q at x in order to universal y are at y this is a preposition why even though q attacks us up doesn't have a quantifier before it but the logic for this part of the equation extension x is shared among p and q so existential for p and q if it's shared it means one value will be substituted for our function if the value of x is one the value here is one and for q it's still one okay so that's the
meaning of shared quantifier okay you could have a shared quantifier or an exclusive quantifier like in the second example the only difference is the value that we are substituting to our function if it's shared one value will be substitute throughout that term and if it's separated we have an exclusive the value for x here is different from the value of x here there is a possibility they are different okay going back to the third example this is a proposition this is true okay this is true proposition rather for example this is not a proposition why
we don't have a quantifier for q at y yes we have an extension x here but it binds x but not y therefore this is not a proposition yes we have universal y here but it is intended for r at y only it is exclusive for r and y it is not shared with q and y so still q advice not bound so therefore this statement is not a proposition okay so now let's move on with predication quantifier specifically negating quantification so let's consider this all student in this class have read here so the negation
of this statement is there exists a student in this class that does not have red hair so it means one of you does not have red hair okay that's the that's the negation of the statement so how do we get this statement okay so we have two ways one is negating the quantifier of our subject or negating the predicate okay so if we're going to negate the the quantifier of the subject this will be our equation not universal x p at x so when we read this we just write not before all so it means
not all students in this class have read here that's the equivalent of this equation or if you want to have to negate the predicate so from this original statement we transfer the negation from left to right okay going to the the proposition function but we need to change the quantifier from universal to existential so the new equation is extension x not be at x and this equation is used to define this statement there exists existential and on the predicate part does not not b so when you indicate a function you negate the predicate okay the
function you negate the predicate so does not have right there not p that's the equivalent of it okay let's have another example there is a student in this class with red hair some one of you some of you few of you have read here so when we negate it the answer would be all student in this class do not have thread here okay so how do we get this sentence so same as with the previous slide we could negate the quantifier or negate the predicate so if we're going to negate a quantifier the equation will
be not off exponential x at p at x so when we read this no student have read here has a red a rather okay no student has a red hair non-existent so if you negate extension non-existing so no none exist none no student have red hair okay we have it here and in case we want to negate the predicate we just transfer the negation from the quantifier going to the purpose proposition function okay from left to right but the existential should be replaced by a universal okay so this is the basis of the answer of
this statement all student universal does not do not have read here negation of function negation function negation of predicate do not have red hair okay so by moving the negation we change the quantifier extension become universal universal become show and we move the project out um the movement okay from left right right to left okay that rule is followed okay so that's it for this video lecture we're going to discuss some sample problems on the next video lecture thank you for listening see you then [Music] you
