WEEK 4 - 4 QUANTIFIERS EXAMPLE PART 2

[Music] hi more example will be discussed in this video lecture let us answer this problem let's see at x be the statement x has a cat b at x with the statement x has a dog and f at x be the statement x has a ferret express each of this statement in terms of c and x d at x and f at x quantifiers and logical connectives that the universe of this course consists of all students in your course so let's start with letter a a student in your class has a cat a dog and

a ferret so for a the quantifier is existential because a student means at least one of the student in this class okay and this existential is shared to all function the reason is we pertain to a student in the whole statement okay so a student means for example joel joel has a cat a dog and a ferret so if we have that scenario it means the quantifier should be shared okay so the answer will be for letter a extension x c at x and b at x and f at x so for b all students

in your class have a cat a dog or a ferret so like in a okay so then the quantifier is shared among the function the only difference between a and b is the quantifier and the function the logical function so the result will be universal x c at x or to d at x or to f at x okay letter c some student in your class has a cat and a ferret but not a dog same as with the prison um example so we're going to have a shared quantifier which is existential okay and the

main function is and about having a dog okay so the result will be extension x c at x and f at x and not the attacks letter d no student in your class has a cat a dog and a ferret so no student means non-existing non-existing means not existential so the quantifier should be negated existential so this is almost letter a but negated quantifier and for letter e the quantifiers existential because for each of the three for each okay so for each of the three animals cat dogs and first there is a student in your

class was one of this animal aspect so the quantifier is also exclusive for each function so that we could have different student for each function okay so we don't have we're not going to share the quantifier but instead an exclusive quantifier for each function so that we could have different values for each function so the result will be extension x c at x and extension x d at x and x succession f at x so it means if this is joel this is grace and this exploded okay we could have different answers different values for

x if we have an exclusive extension for each function okay so that's problem number one let's move on to the next sample problem so let q and x be the statement x plus 1 greater than 2x if the universe of this course consists of all integers what are the truth values so we're going to determine if the given values will make the statement true a q at zero so substitute zero plus one greater than two times zero definitely true q at negative one negative one plus one greater than two times negative one definitely true because

zero is greater than negative two q at one one plus one greater than two times one it's false because they are equal letter d extension x q at x is there a value that will make the statement true yes based on a and b so this is true letter e all values of x will make the statement true this is false based on c because there is a value of integer will make the statement false so that's why letter e is false f there is a value of x that will make the statement false yes

this is true based on c there are values that will make q at x false like number one value of one rather and letter g are the uh all values of x will make the statement false definitely false because based on a and b it is true g is false okay so that will be our answer for this sample problem next question so in the question below suppose the variable x represents students and y represents courses so you at y y is an upper level course m at y y is a math course f at

x x is a freshman a at x x is a part-time student t at x y student x is taking course y so write the statement using these predicates and need and any needed quantifiers so in this example we have two groups so based on the given we have student and course x for student y for course and one is for subject and one is for the predicate okay in case we have this scenario we need to create a function that will connect this to group that belongs to subject and predicate so through this function

t at x y so true t at x y student is connected to course y okay again if we have a subject if we have a group in our subject and another group in our in our predicate we need to create a function that connect these two like p at x y okay student x group one is taking course y group two okay so let's convert the first sentence so every student is taking at least one course so no specific student and class for this statement just a general student and general course so therefore we're

just going to use the x y to convert this into an equation so the quantifier is for the subject is every it means universal at least one existential so the answer will be universal x extension y t at x y next example there is a part-time student who is not taking any math course so in this statement we have a specific student which is being a part-time student and a specific course which is math okay and the quantifier for the student is existential so to connect the predicate to subject to subject to predicate we need

to accept why and to determine the the specifics about the subject and predicate we need to write it before the function that connects these two two groups okay so part-time student followed by math course and then connecting function with this t at x y that will be the pattern on how we're going to to answer this okay so the answer will be extension x there is a part-time student extension x universal y n e so it means all so that's why this is universal y okay part time a at x this is our subject a

at x and to connect our subject to our predicate check our quantifier for x extension so therefore for the next quantifier for the next function to connect to the next function we're going to use add extension and okay after and this is all part of the predicate so what is our predicate math course so emma why okay and then implies why it lies because universal okay universal means connecting my to the next function through implication and then negate the at x y so why why do we put negation at the x why not in math

so go back to our sentence not taking not taking is part of what function is it math or is it t at x y it is part of the x y taking course so we are going to negate the taking course so not taking means not negation of t not of the other okay so that's why we put negation here not on m at y because we're navigating math but we are negating that taking so that's why we are negating t so this is our answer for letter for this sentence extension x universal y a

at x and m at y implies nothing at x y and also to separate the subject to the other function we need to group to have a separate parenthesis for the subject and parenthesis to other functions after the subject like this one okay m at y is related to not the other x y okay because if we're not going to group it we're going to evaluate a attacks with m at y which is incorrect okay so all predicate part all subject will be true okay by a parenthesis then afterwards the rest will be grouped in

another parenthesis okay another example every part-time freshman is taking some of upper-level course so we have two specifics for our student being a part-time being a freshman and for our course upper level and the quantifier for for the student for all universal for of course some existential so the answer will be universal x extension y okay for the quantifier of x and y and then define the subject being a part time refreshment so f at x and a at x this should be group s1 since this is our subject and connect it with the predicate

and the rest through existential because the quantifier for x is universal so to connect the subject to the next function implication again because of universal and then everything after the subject will be group as one so the specific for the course is upper level so you add y and that to y and because this is extension connecting u and y to the next function is ant based on existential and t at x y so this is our answer for this sentence okay another example so from a mathematical equation transform it to an english statement so

we have four statement here m at y y is a math course b at x x is a full time student f at x x is a freshman t at x y x is taking y again because we have two groups one is for the subject another is for the predicate x is for being a student y is being four is four courses okay so we have universal x expansion y d at x y so we don't have specific for subject for our subject for our student and for our course so just quantifier so the

answer will be every student person is taking a course of course means extension okay next extension x universal y t at x y so now we have a different quantifier so extension for x universal for y so the result will be some student is taking every course some extension every universal next example universal x extension y we at x and f at x implies m at y and d at x p at x y so quantifier for universal x for x is universal for y is extension we have two specifics being a full-time and being

a freshman so full-time freshman mli math course so the answer will be every full-time freshman is taking a math course every universal full-time freshman b at x f at x taking the attacks my course met y extension that's why this is just okay so that's our answer now we're going to negate the given sentences okay so we're going to indicate what is the negation of some bananas are yellow we have two answers no bananas are yellow when we negate the quantifier sum negated non-existing so no bananas are yellow or second answer all bananas are not

yellow so some will become all and then the gate the predicate are not yellow so that will be our second answer another answer all integers ending in the digit 7 are at first answer negate the quantifier not all integers ending in the digit 7 are add second answer negate the predicate so all will become sum so the answer will be some integers ending in the dg7 are not at from all negation go into the predicate so that's why it becomes are not odd okay third no tests are easy no test means negation of existential so

when we negate non-existing so it means it exists so the result will be some tests are ec fourth roses are red and violets are blue so we're going to use de morgan's law here okay from a shared negation it will become roses are not red or violets are not blue some skiers do not speak swedish negate a quantifier no skiers do not speak swedish or transfer dedication from no skiers all skiers cancer the double location all scares pick swedish okay so when we transfer the negations from some to the predicate we have a double negation

so we could cancel the double location so that's why we have speak swedish as predicate okay another example so we have a statement f at x x is a freshman a at x is the part-time student d at x y x is taking y we're going to substitute the values given the specific values given on the on the equations to our sentences okay f at mikko f so mikko is a freshman not extension y p joe y so joe is a specific value y is bound by not extension y okay so the answer is do

is not taking any course so how how did we come up with this answer so we transferred the notation from existential going to the function t so that's why it becomes is not taking any course but if you're going to use static special why it's okay girl is taking no course if you want that it's okay just taking no course or joe is not taking any course that will be our answer for this and last extension x a at x and not f at x so our subject is part-time student and being a freshman okay

the answer will be some part-time students are not freshmen okay next example we're going to determine if the uh what is the equivalent of the given equation based on the three english statements okay we're in effort x x is a freshman m at x x summit major where in x represents student so the first statement could be some freshman or math major second every math major is a freshman third no math major is a freshman so let's start with the first given universal x and m at x implies not f at x so initially the

sentence is equivalent to all math major is not a freshman or are not a freshman okay but it's not equivalent either of the three so we need to do something with our equation so we could transfer the negation from right to left or used capacity to rewrite it so everything that we learned about implications could be implemented to solve this problem okay but for me my solution will be from predicate going to our quantifier of our subject that's how i'm going to answer this okay from not f at x transfer this negation go into the

quantifier universal x and in effect universal will become existential so this will be our equation not extension x m at x and delta f at x y and because this is already existential so that's why we need to change the function and how do we read this in english no okay non-existent it means none no math major is a freshman so therefore this is equivalent to three this equation is same as three okay by manipulating the equation we could and we found out it is a it is the same as number three no matt major

is a freshman okay another example not of extension x m at x and not f at x so no math major are not a freshman or not no math major is a it's not a freshman okay so it's almost the same as three but the problem is the predicate is negated so therefore it's not three okay so why not transfer the negation from existential going to the predicate so extension will become universal double deviation cancel the negation so we have this cancel so we have universal x and x implies f at x so what's the

equivalent of this every math major is a freshman equivalent to two okay so we move the negation from left to right and double negation hola we have number two the equivalent of this statement is equivalent to second statement okay another example universal x f x implies not m at x every freshman is not a math major it's not equivalent to one two three initially so we could transfer negation from right to left or for me instead of transferring the negation i'm going to use contrapositively write this so the equivalent incontropacity of f attacks implies that

not m at x is universal x m at x implies not f at x if still remember the first given in this problem this equation is the same as that equation so therefore it's equivalent to three okay so by moving the negation by transforming it to a different form even though when we transform it into english statement okay we'll have a different pattern but they have the same meaning okay so that's it for this video lecture so thank you for listening see you on the next video lecture thank you [Music] you