00:00 Good day, my name is Afeez. I will be your further maths teacher for today. This topic is

00:18 sets. At the end of this topic, a learner should be able to know the idea of a set, set

00:26 notations, applications, disjoint sets, and vendor-gram. So let's look at the definition of a set. A set

00:36 can be defined as a group or a collection of where defined objects or numbers or elements. So

00:42 example is collection of books, cooking utensils. A set is denoted by capital letters such as p, q,

00:52 and h, e, c. Why small letters are used to denote an element of a set? So let's define what the

01:04 element of a set is. These are the elements or members of a set given and also the element are

01:14 separated using commas and enclosed, called a curly brackets. Then we have example of element of a

01:24 set. If we have M, which is a set, one three five seven and eleven are the elements in that set M.

01:35 So example, if we have to write down the element in each of the following sets and we have

01:43 a, f, and m. A defines the set to be odd numbers from one to 21 and f defines the factors of 30 and

01:54 hem defines multiples of four from up to 40. So we have the sets A, which is the set of odd numbers

02:05 from one to 21. So we have one three five seven nine eleven thirteen fifteen seventeen nineteen

02:12 and twenty one. Why we have the set f to be one three five sixteen fifteen and thirty and those are

02:22 the factors of 30. We also have M, which is the set of, which is the set of the multiples of four

02:33 from the multiples of four up to 40. So we have four eight twelve sixteen twenty twenty four

02:42 twenty thirty two thirty six and forty. And also, let's also define the cardinality of the sets.

02:48 When we say that it said, it said to be having a cardina that is if the number of elements in the

02:57 sets, that is, we have, if for example, we have a set A and let's set A contains one three and five.

03:07 The cardinality of the set is denoted by this and this is the number of elements in the set A.

03:16 You know, we can see that we have many number one three five and that is three. So this shows that

03:22 the number of elements in the set A is three.

03:31 Example number one, given that the universal sets equals all the days of the week B sets

03:41 equals all the days of the week whose letters begins with S. And we have to list all the elements

03:48 of the universal sets. We have to list the members of B and we have to find the number of

03:56 universal sets then we are also to find the number of universal sets plus the number of a set B.

04:03 So let's list the universal set element first. We have

04:09 and that is all the days of the week. Many that I will use the first letter to denote that

04:20 M, which is the Monday, start from Monday. We have Tuesday, T, we have Wednesday, W, we have Thursday,

04:29 T, H, we have Friday, F, we have of Saturday, S and we have Sunday, S. That is the

04:40 elements in the universal sets. Also, list elements of the members or elements of B.

04:47 And what is it said all days of the week whose letter begins with S. We only have two

04:56 sets. I mean two days of the week whose letters start from S, we have S, which is Saturday and Sunday.

05:05 Then also what is the cardinality of universal sets? That's the first thing.

05:15 I mean that the number of elements in the universal sets. So we have one, two, three, four, five,

05:20 six, seven and that is seven. Also, we have to find the, or we have to find the cardinality of

05:29 universal sets plus that of B. And how do we do this? Our B is one. We have the cardinality of B to

05:40 B to. And we have that of a, this to be seven. So it means we have to have seven plus two.

05:51 And that's give us nine. Also, let's look at the example number two as well.

06:07 Example number two. We are to list, list the members or elements of the following sets.

06:18 Our A is x such that five is less than x or less than eight. And it means our A, we start from

06:33 six. Five will not be included because F, I mean five is less than x. So six, seven,

06:42 so we also, our eight will also be, we not be included. We also have B,

06:48 which is a x such that zero less than or equal to x less than or equal to five.

06:56 So it means we will start listing the element from zero. One, two, three, four, five because zero and

07:07 five will be included because it is less than or equal to. So these are the elements of A and B.

07:14 So let's also take a look to the type of sets. We have binite, in finite sets. And what is a

07:25 finite set? When you are able to list all the elements or members of a particular set,

07:32 then we call it what? We call it a finite set. Also, we have an infinite set. When you are

07:38 unable to list all the members, for example, if you have a finite set, set A is the finite set,

07:48 because we are able to list the elements. If for example, what are the infinite sets? If you are

07:54 unable to list the elements in the particular set. If you say the number of people with one

08:06 of two eggs, you know, we cannot list those people with two eggs. We don't have anybody within the

08:12 planet with two eggs. So that is that about the type of set, which are binite and infinite sets.

08:19 Also, we also have an empty set or we call it a non-set. A set without any member or element,

08:27 it is called an empty set. And let's say we have this.

08:36 Let's say we have if, for example, the set A is this. This is an empty set or we use

08:52 this to denote our empty set. So that is an empty set. Let's also look at the type of set,

09:02 which is a subset and supersets. If, for example, we have a two sets such that all the elements in A

09:11 is containing B, then we set A is a subset of B. If, for example, we have A equals one three five,

09:21 and we have our B to B. Let's say we have two three five. If you look at these elements,

09:31 there are some elements in A that are also in B. If A contains at least two or three of the

09:40 elements, then we say A is a subset of what? It is a subset of B. Then we also have a superset.

09:48 We use this to denote a subset. A is a subset of B. This is what we use this to denote. That is,

09:57 A is contained in B. We also have a superset. We use this to denote a superset. If we have B,

10:05 is a superset of what? Of A. So we use this to denote. You can see the difference between this

10:11 is subset and a superset. That is, if there's at least one of the one elements in a set B,

10:19 but not in A. So it means there's element in a set B. If you look at this value,

10:24 there's element in a set B that is not in A. So it means B is a superset of what? Of A. Do we

10:31 understand it at all? So let's take a look to the next slide, which is a disjoint set.

10:38 Two sets are set to be disjoint. When there is no element in common. If for example,

10:44 for example, we have A equals to three and we have B to be one and five. You can see there's

11:06 nothing common between A and B. So this is an example of what? It's an example of a disjoint

11:13 set. So let's look at the next slide. And that is the diagram, the Venn diagram of the disjoint set.

11:20 You can see that there's anything in between A and B. And that is a universal set. You know,

11:26 we set earlier that we use, we use, we use these to denote a universal set. So let's define what

11:36 in the first set is that a set that contains all the members under the consideration of any given

11:44 problem. That is, if you have A, which is a set and we have also B, which is also set.

11:51 The members are elements in A and B. If we take them together, they are universal sets.

12:01 We can also have C. Our set can also be more than two or three or four. So all the elements

12:08 in A, B, C without repeating the number or the members of the element in the sets,

12:14 they are what? They are universal sets. And we use these to denote it and we use all we use

12:20 you to denote it. Complementary sets. This is a set that contains the members in the universal

12:27 sets that are not in the sets A. If, for example, we have universal sets to be this,

12:33 let's say we have one, two, three, four, and five. And we have A to be, let's say we have

12:48 six, seven, eight. You can see that the complemented sets, you know, we said that contains the members

12:59 in the universal set that are not in A. Then we use these to denote it. A prime or A complement.

13:09 So if you look at this now, universal sets, one, two, three, four, five, and we have a set, six, seven,

13:18 eight. If you look at this now, you see that there is no element that is universal set of,

13:25 there's no element in A. Or let me say, members in the universal sets that are not in A.

13:35 We have one, two, three, four, and five. You can see this. So those elements that are in universal

13:47 sets, but they are not in A at this. So these are the complement or, I mean, these are the

13:54 complement sets, or we call it the complemented sets. So let's look at the intersection of sets.

14:09 This is a set which consists of the common elements in a given two or more sets. We use these to

14:18 denote it. That is, we use these to denote it. And if for example, how do we find this? Let's say we have

14:24 A equals one, three, and five. And we have B to B, two, three, and five. And we have to find the

14:37 intersection of these two sets. So we set A intersection B. And what is the element that's

14:46 common between A and B? We can see that three is common. So we have three. So that is an intersection

14:53 of two sets or two or more sets. Also, you know of a set. This is a set of all members that

15:01 belong to A or B or to both elements. If for example, we have A equals one, three, five. We have B

15:11 equals two, three, five. You see that we use this to denote the union of a set of two sets.

15:19 So how do we find the union of these two sets? It means we write all the elements in A and in B

15:27 together. Without repeating those ones that are common in between A and B. So we have one,

15:36 two, three, and five. So these are the what these are the union of A and B. So let's look at the next

15:44 slide. So let's look at the relationship between union and intersection sets.

15:58 So let's say we have what is the reason between these two? We use what we call a cardinality of

16:09 a set to do that. A or B, that is equals the number of A plus the number of B. Then we have

16:21 minus n the number of this. Or we can say the number of that is the cardinality of A union B

16:34 we give us this. The cardinality of A plus the cardinality of B then minus the cardinality of A

16:46 intersection B. So these two, is that how you write it in this form or you write it in this form?

16:52 When you have A or B it means union. If we have A and B it means it means if we have it means

17:03 intersection, it means intersection. If we have A or B it means union, if we have A and B it means

17:14 intersection. So now let's look at the example. We want to use this example to differentiate between

17:26 the universal set, I mean to differentiate between intersection and union. So if we have,

17:33 let's say we have the number of A, don't forget we said the cardinality of A says can be represented

17:41 by this. So we have 23 and we have the number of B to be 12 and we have the number of A union B

17:51 to be 35 and don't forget the previous formula. In our previous slide that says the number of A union

18:03 B equals the number of A plus the number of B minus the number of A intersection B and what is

18:14 our A, the number of A union B is given to be 35 and the cardinality of A is given to be 23

18:26 then plus we have the cardinality of B to be 12 and we have this to be and we have to find this.

18:37 So we have to find this. So what we see is that we have 35 equals 23 plus 12 we have 35

18:51 then minus minus this. So we have, so we collected like times we have the number of A intersection B

19:12 equals 35 minus 35. So and the number of A intersection B would give us zero. What does that mean?

19:21 It means this is the disjoint sets and if you look at it we said sets A and B are what they

19:28 are disjoint sets. In our previous definition you know we define what disjoint set is. So in conclusion

19:37 this is an evolution of what we have done so far we have universal sets X such that so X one

19:48 less than or equal to X less than or equal to 12 and the subset of those university's D, F and G

19:55 are given as follows. D equals X such that two less than X less than it's then half or so as well

20:04 and we have G and we have to find A or D union F we have to find D intersection F we have to find

20:11 D intersection G complement and we have to find G complement. So the int has been given also

20:17 least the element of universal sets, the set D, the set F and the set G.

20:24 Thanks for watching. Once again my name is Mr. Fees, see you next class.