[Music] in the previous example we only have one subject per preposition so how can we express if the subject is a representative of a group or the whole group itself so for example every computer inside it is protected by intrusion detection system so in this example the subject is computer and the predicate is all about protected by intrusion detection system so how could we um state it in a mathematical equation wherein this statement is true to all or for all computers of saw it okay another example there exists at least one student who has a
pink here so in this statement in this sentence the subject is student and the predicate is having a pink hair so how could we represent or how how could we convey this statement this logic in our equation so to do that we need to follow some steps okay so first let's have this example x is greater than 10. so the first step is to define your group or the member or the subject of the statement so in this example it is x so x could be a number from a certain group okay so x is
our subject then second step determine the predicate which is is greater than 10. then from there we're going to create a prepositional function third step so how do we create propositional function so we use capital letter to represent a function so for example p and then followed by a parenthesis and inside the parenthesis we have the subject or the variable that represent the subject in this example it is x okay so this is a simple propositional function so what is a proposition function it defines the subject and the facts about it so p at x
is equal to x is greater than 10 okay so if we know the value of x just apply it to our equation so when we evaluate it we know how to evaluate the equation so for example x is one so substitute to our equation one is greater than ten it's false am i right or 11 x is equal to 11. 11 is greater than 10 definitely it's true so by using propositional function we could use it um to evaluate different values of x okay so that's a propositional function so by the way when we read
this it should be p at x okay it means function at the value of x when we evaluate the the statement okay so let's have another example consider p at x is equal to x less than 10. can we determine if it's true or false definitely not because we don't know the value of x we can save it's true we're going to say it's false so therefore when we don't know the value of propositional function therefore it is not a proposition it will become a preposition if you're going to supply a value so for example
p at one so when we evaluate the equation one less than ten it is true or another example p a 10 10 is less than 10 definitely incorrect why they are equal okay so again after creating a proposition function is that a process proposition function we need to supply a value so that it will become a proposition so that we could determine if it's true or false another example let's be at x x is a multiple of four initially we don't know if it's true or false we need to supply a value for x so
for example two six seven okay so by supplying a value we could determine if it's true or false okay how about this let's be at x is equal is equal to x plus seven definitely no then let's apply a value is it a proposition definitely no because it's an operation we don't determine if it's true or false we just do the operation to determine the result we don't compare so therefore even though we know the value of x in x plus seven it's still not the function okay unless if we uh um related to another
value okay anatomy of our position function again we need to define the variable that represents the group the member of the group the predicate is the topic about that subject and the function itself so we use capital letter to represent the function okay let's have another example so like in single variable or single subject proposition single proposition we could have multiple proposition in a statement okay so we could have two variables three variables in a statement so p at x y for the first example so how do we evaluate this when we supply value so
the order of xy determine how we're going to substitute our values in our equation if x is the first variable so the first value is for x second variable is y so therefore the second value is for y so when we substitute it in our equation one plus two is equal to zero definitely it's false okay another values are one negative one so one is for x negative one for y so when we add these two it is equivalent to zero so therefore it is a true statement another example three variables p at x y
z again the order determine how we're going to substitute the values to our equation x plus y is equal to z so the first two is for x and y respectively 5 is for z 3 plus 4 is equal to 5 false how about p 1 2 3 1 plus 2 is equal to 3 definitely it's true again the order of variable determine how we're going to substitute the defined values in our propositional function okay now in a proposition function there are instances we can't substitute all values or we don't know the values that will
determine whether the statements true or false like in our first few slides talk about for all computers or or um there is a student that's uh that has us pink hair but we don't know who was the student so instead of specifying the specific value we could use a quantifier okay so quantifier specify if the statement is for all or for a certain member of a group okay so we have two types of quantifier universal and existential so let's discuss first universal this is the symbol for universal absolute upside down capital a okay and when
we write it in our equation it should be written before the variable okay before the function so universal xp at x so quantifier the variable that we are talking about and the function okay so it means for all values of x p at x is true okay that's the meaning of it okay so let's evaluate x plus one greater than x so is this true definitely i don't know why because we don't know the domain of the group yes there is a guest domain it probably it is probably a numerical value but we didn't declare
it so therefore you could not assume that it is a number or a numerical value it could be a character it could be a name okay so for example a if it's a a plus one is greater than a definitely no or a state minnesota plus one is greater than minnesota definitely no so when we evaluate or when we include universal quantifier we need to define the domain or the universe of this course okay to determine how we're going to evaluate the equation so if it's an integer so we could say it's true or false
okay so is there a value in an integer when we add one okay it will be greater than the the original value definitely yes okay another example let p at x is equal to x divided by 4 is less than x let the universe be a real number real number it could be a fraction a whole number a negative number a positive number etc okay so is this true definitely it's false why pick true negative numbers negative numbers is part of real number okay so negative 1 4 for example x is 1 negative 1 is
less than negative 1 definitely no negative 1 4 is greater than 1 okay therefore this statement is false it is not true to all real numbers so when we change the domain so for example instead of real number we make it positive integer that's the time the statement will be true because all positive integer when we supply it when we substitute it to our equation the statement will be true so therefore it will be true to all okay universal statement will be true if and only if it's true to all values or true to all
members of the group if not the statement is false like in this example okay and again by changing the group may change the status of our statement it could change to from false to true or true to true or retain the value so it means when you are proving something you can compare two groups okay by stating the same facts so for example group a group b so group a you have this facts and you want to apply it to group b okay it doesn't follow you need to throw a bit you need to do
experiment first or you need to do studies before you could say it now this is true for b group b okay so again if it's true to a certain group it doesn't follow if it's true to another group okay so based on this example now let's move on so for the universal quantification the logic behind it is and since it should be true to all members of the group so that's and definition okay now let's move on with existential quantification if say existential it means it exists it is present in our world okay so it
is represented by backward e and same as with universal it should be written before the variable that represents the subject followed by the propositional function okay so let p at x is equal to x plus one greater than x so when we use existential it will be like this existential x followed by p at x so what does it mean there is a value of x that will make the statement true that will make p at x is true we don't care if it's if it's if or if if all values will make the statement
true okay we only care there is there is an existing okay we don't care if it's true to all if there is one the statement will be true for existential okay so for example x plus one is less than x okay so again we need to specify the universe x is a real number is there a value when we add one it will be less than x definitely none am i right you add one so definitely it will be higher than the previous uh the original value so this statement will be false so if it's
non-existing access style our existential statement is false non-existing means existential quantification is false there is no value that make the statement true therefore the statement is false under existential okay another example x plus 1 greater than x again we'll talk about integer or or even real number so when we when we substitute the value here from that group the statement will be true regardless if it's all okay again we're talking about there exists a value there is a value right there okay at least one member of the group that will make the statement true existential
statement is true okay don't forget that so if for universal it is based on and existential is based on or so at least one true the statement is true one member make the statement true the statement is true same logic with or operation so that's existential okay so as a summary so we define p at x as a propositional function we use capital p to represent the function and the equivalent of the p at x is the subject equal to predicate okay or the facts about the x then proposition function is not a proposition if
we don't know the value of our variables so we have two ways to determine the value so by supplying the specific value like five and zero or provide a quantification will it be universal or existential okay so that's it for this video lecture on the next video lecture we're going to discuss some examples about quantifier and predicate thank you for listening see you on the next feature lecture [Music] you
