Waching daily Aug 30 2018

okay guys so today what we're going to look at is intersection and unions but

first in order to understand that we have to understand basic concept in sets

which is Venn diagrams so this is your first Venn diagram and a Venn

diagram basically shows relationships between different sets so in this case

we have a diagram that shows a relationship of people who like ice

cream and people who don't like ice cream specifically grade 10 teachers so

as you can see teachers that like ice cream mr. David myself mr. Madjid mr.

Abel mr. Stanly and mr. Matt so these are people who like ice cream in our

part of the set a of people who like ice cream

over here we have A complement which is basically not A okay we

can call A complement that's the same as not A and that basically means people

who don't like ice cream right people were outside of this Circle so in this

case I put mr. Sharif there and Miss charlie I don't really know it these

people like the history or not I just sort of made it up for this example but

there we go so the first property of Venn diagrams is that if we need to find

e or epsilon as the universal set okay and in this case the universal set is

all grade 10 teachers if we define that as universal set the number of A plus

the number of A complement must be equal to the total number of population within

the universal set so in this case the number of a is 5

And the number of A complement is 2 and a number of people within this whole

box this whole Venn diagram is that is 7 okay so now let's

take a look at the idea of intersection so the definition of intersection is

this little upside down to you and if we have the definition let's try this way

okay so now let's take a look at the idea of intersection and the definition

of intersection is E intersects H where all of X's and X is a member of E and X

is a member of H so we would basically say E intersect H and basically it's just

"AND" right? the key word here is "AND" we're looking for members of both sets

that intersect right that's what near section means in this case if I give an

example E is the set of teachers who like ice cream

okay let's say that's the same as last examples of mr. Madjid, myself mr. Abel

mr. Stanly and mr. Matt and H is a set of teachers who like pizza mr. David mr.

Stanley mr. Abel and mr. Richard we could create this in a Venn diagram and

the place where both circles overlap is the intersection so let's look at which

teachers overlap well we have myself I like ice cream and pizza mr. Stanley

apparently likes ice cream and pizza and also mr. Abel seems to like ice cream

and pizza so we could put them in this intersection point here

mr. David mr. Abel mr. Stanly

now outside of this intersection point would be teachers that for example only

like ice cream or teachers that only like pizza

so which teachers only like pizza? well according to this mr. Richard only

likes pizza but he does not like ice cream and according to set E it looks

like mr. Madjid and mr. Matt only like ice cream but don't like pizza

and keeping in mind we can also still include teachers who don't like any of

these right there are some teachers who don't like ice cream or pizza for

example maybe miss charlie and so maybe miss

charlie doesn't like ice cream or pizza we would

write her outside of the Venn diagram but still part of the universal set

which is grade 10 teachers okay so continuing our examples now we're going

to take a look at the idea of Union so this looks a lot like intersection the

thing that we just looked at but this is a little bit different instead of the

upside down U this is a right-side up U so this we call E Union H and

the key for union is that instead of X is a member of E and extra member of age

we just say x is a member of E OR x is a member of H so as long as they are a member

of one set or the other or both they are considered part of this set called E

Union H so in this case the teachers who like either ice cream or pizza would be

basically all the teachers inside the circle right? mr. Madjid, Mr. Matt

mr. David mr. Abel mr. Stanly and mr. Richard they're all part of the set a

union thing now doesn't matter that mr. David mr. Abel mr. Stanly liked both

pizza and ice cream no doesn't matter right because they're still considered

part of the set that includes before does it matter that they're in both sets

so should we list them twice no not really

right we still only listen you know mr. David mr. Abel, mr. Stanly only one time

we don't put them twice in the same as long as

they are part of one of the sets or part of both sets, they are considered part

of the group. Quick check if I said E Union H complement right not E Union H what

would that be take a minute and pause the video

okay so in the example E union H complement as in not E union H well that basically

says everything that's not inside of E or H so in this case all we have

outside of the bubbles are Miss Charlie right so this set

e union h complement would be Miss Charlie

okay guys so now let's take a look at a couple of more properties that relate to

sets in this case we want to talk about the idea of disjoint and the definition

of the disjoint is when we have two sets A and B that when they intersect they

form the null set and remember the null set the symbol null set just means that

there's nothing inside that set so this is an example of a disjoint situation if

we have a set of a which has the numbers one two and five and the set of B which

has the numbers three six eight and nine there is no overlap between A and B

right there's no number that is in both set A and set B so we would call these

two sets disjoint now also in general and this is a general property that

applies to all sets not just this example A union A complement is going

to equal the universal set and if you think about that basically makes sense

right everything inside A plus everything that's not inside A all

this stuff outside should equal all the stuff within this Universal set so A

union not A should equal the universal set and by the flipside A intersection

not A should be equal to null set right if I say everything inside of a and

compare that with everything outside of a is there any intersection well no

right because by definition everything outside of a is outside of a that's what

A complement means and by definition everything

inside of A is only inside of it so how could we possibly have an intersection

between A and A complement so by definition it must be the null set there's nothing that intersects between the two sets

okay guys let's take a look at example number 17 which is on page 14 of your

book and in that example we have a universal set which is the letters P all

the way to W and we have a set A which is PQRS we have set B rst and we have

set C which is STUVW question asks us to find the number of elements within A

union B and also to list the elements of A union B complement and A union B Union C

so let's take a look at it first by drawing a Venn diagram and when I draw

a Venn diagram what I want to do is I basically go through all the elements

and I see which sets they are apart of so in this case element by element let's

start with P well P looks like it's only part of A so let's put P in A

Let's look at Q - Q is part of A and nothing else

we've got R and R is part of A it's part of B also so it should be here

okay what about S? S we have in A we have in B and we have in C so that

should be in the middle shared by all three sets

okay T - T is part of B and part of C put T over here. U - what about U? the

U is part of only C. V is part of only C as well and W is also part of C so

in this example first thing we want to find is the number of elements within

A union B remember N() means the number right not what is in A union B but

the number of elements within it so what is in a or b we have p q r s t that is

one two three four five elements so this would be be five but if we asked you

for the elements of A union B the question was just this A union B then you would

have to actually list out p q r s t but in this case it's asking for the number

of elements okay secondly we're asking us to list the elements and this is

where we actually have to write out what is there so A union B complement a union

B complement so that means A union B is these two guys and everything that's outside

of it so what is outside of A or B well that's these three guys over here

W U and V. So A union B complement is U, V, W

and what about A union B Union C well that's either A or B or C so it's

everything that's inside these three bubbles in that case every single

element right the entire Universal set basically the universal set. okay guys I

know you're tired but this is the last example and this is an example seven

from your book if you take a look at the problem it says that the universal set

is basically X where X is all the numbers between 2 and 30 and X is a

member of the positive integers that's Z-plus so X is a positive integer between 2

and 30 that basically means X is 2 3 4 5 6 7 8 9 all the way to 30 not 2.5 not 6.8

that's just whole numbers between 2 and 30 now we have a set P where

consists of X and X is a factor of 30 and the question asks us to

list the elements of the P now keeping in mind that X must be between 2 and 30

so if we begin we must know that one cannot be part of the P because 1 is

less than 2 so the first member of P must be 2 - and what else is a factor of

30 well 3 is and we have 5 we have 6 10 15 and we should have 30

itself this is less than or equal to including 30 and 2. The second part asks us to

describe the set P complement so P complement is basically everything

outside of P but still part of the universal set everything that's part of

the universal set outside of P and we would write it like this

so X where X is not a factor of 30 so that would include numbers such as four

there's no four seven eight nine those are not factors of 30 all right finally

we want to find the number of P complement so what are the number of

elements within this set P complement? that's what this N means so we could

find that by subtracting the number of Universal set and the number of P set

N(P complement) should equal the number of elements in the universal set

minus the number of elements in the P set

that's one of our properties so in this case if we take the number of elements

within the universal set that's count them two three four five six - if we count them

up we would find that the number of elements in there is actually you can

find the number of elements for set P well one two three four five six seven

or answer 29-7=22 okay thanks for all your hard work I will not give you a practice

problem because this was quite a long video so I'll see you guys