SETS class XI lesson 01

003                            TOPIC: SETS                          class XI                      lesson No: 01

** What are sets?
You must have heard of a childhood puzzle, “ katore mein katora…”
And that song, “ meri duniya hai maan tere aanchal mein…”. So small is the lap of Mother yet it is the World for all of us.
And teacher telling, “ in a class of 100 students, 90 speak Hindi,84 speak English, 45 speak Tamil, 64 speak Punjabi, and so on…”. You wonder ,” yeh class hai ki water-of- India, khatm hi nahin hota.”
Then there are double roles in films. List is almost endless.

Keep the above in mind.

** A SET is a collection/ group of known number of SIMILAR things taken together. Each thing must belong to a Class yet each is UNIQUELY identifiable. Like, a class in school has students. Students have names. Student can be boys or girls.

**To make each student UNIQUE each student has a roll number. Obviously two students cannot have same roll number, they may be having same names as Ram, that is why we assigned roll numbers to them.

**There can be only limited number of students in a class which we can COUNT.

Let us call them as members of the classX in school. OR students are members of a SET called classX.
Next time we could take a group of numbers like 1,3,5,7,9 OR all prime numbers less than 100.

** A set is like a “pudiya” (packet) that we make to keep some things together.
Now within a pudiya named classX, we can make two sub-groups of boys and girls. That is, two pudiyas inside main bigger pudiya called classX. Hence, Katore mein katora…
We can show pudiya as: { }
Members as : Ram, shyam,sita,Gita
Put them in a pudiya as : { Ram,Shyam,Sita,Gita}.

**This is a set having four members which are all DIFFERENT from each other.
Suppose we say, set of all the natural numbers, OR set of all even numbers, OR set of all beautiful women, Or set of all rich people. Can you define such a set?
No, we cannot. Why?
Because, we are not able to define in mathematical terms how we shall decide what is a rich person; where all the natural numbers will stop etc.

Our set must be countable and strictly defined. Like a man is rich who has an i-phone or one crore rupees etc. That is give definition of “RICH”

** You see one person (a member of set) can be a man named Ram, who can have so many characteristics: he is less than 20 yrs old; can speak English and Russian; is a brother of Hema; is rich, is a Punjabi; is 5 feet tall etc. Ek anaar aur sow bimaar.

This tells us that given a number of students, say, 10 students, we can form so many different sets based on what we are interested in like: set of rich students; set of English speaking students; set of North Indian students and so on.
Now write down some examples of SET.

Remember: you should be able to SPEAK out what you mean. Set is all about COUNTING.

** If you are talking about your school that has 3 class rooms of class X, XI and XII and school has 10 staff in office and 20 teachers then this set of (10 + 20 + total number of students in all three classes) is our DUNIYA for this particular problem.
In the above student example our duniya consists of 10 students. We do not look at anything else for the problem at hand.

We can generate problems based only on members of our duniya. Therefore it is imperative that DUNIYA is clearly defined and understood.
We can define our duniya for each different problem at hand.

** We can also define our criterion based on students speaking some language, say, Hindi. You might say what about girl students or Tamil- see they are not in our criterion. So we dont look at them.  ( katore mein katora)

** Remember this: while a member can have many characteristics but we are concerned ONLY that on which our search or problem at hand is based, we are not concerned about any other. Like if we want to deal with students who speak Hindi and Tamil we are not bothered if they can speak Punjabi also. And our Duniya remains fixed viz our school.

** This duniya is called UNIVERSE in language of Set. Every group inside this Universe a set, or a subset.

**SETS are all about counting.
Speak out what you mean and write in a language others understand without having to ask you.

** Put it in some standard notations.
Example: we can say:
A group of students named R,S,T,P,Q,Y,J when taken as a set can be written as;
Set called students = { R,S,T,P,Q,Y,J} : note we have to name every student. (here total =6)
If a class has two students named R then for sets we take only one R and the set is written as:
{ R,S,T,P,R,Q,Y,J} ie set of 7 students becomes = { R,S,T,P,Q,Y,J}  ie if a class of 40 students has two students named Ram then for the purpose of set questions class is taken as 39 students. Note that here since each student has a different name ALL names have to be written.

** But suppose we tell in a way that every one understands then our writing work becomes easy, like:
A set of all even numbers AND less than 100.
Note two things: all know what an even number is; also, to make the set countable (FINITE) we place a limit on SIZE (here 100).

**One more thing: what is the TYPE of member? Like a number can be: natural, real, integer etc. Or a member can be an animal having 4 legs.

We say a member BELONGS TO — a TYPE
Set A= { x, x belongs to TYPE, LIMIT or range}
A = { n: n ϵ N, 1<N<99}
Means that set A is a set of numbers, numbers are of type Natural, and lie between 1 and 99. We save space and effort. And all understand this.
**S = { x: x ϵ girls, All girls under 13 yrs of age } IS THIS CORRECT? NO, WHY?
S = { x: x ϵ girls, All girls under 13 yrs of age in a class of 45 students } Now it is correct. WHY?

**THIS statement reads as : x variable name BELONGS to TYPE girl, CRITERION: under 13 yrs; FINITE size of set: class of 45 students.
This is beginning of chapter on SETS for class XI.

**Now you can say we have two ways of writing a set:
1. In which there is no pattern or known characteristic. If we make set of such members then we must write each and all members separated by “commas”. This method is called ROSTER.
2. In which there is a pattern like all male, even numbers, vowels, students etc. In these cases we all know. Here we can define the property like : student in a class of 40 students and coming from North India. Then we need not write all the members. This method is called SET BUILDER. Find out which example above belongs to which type of method.