sets class XI lesson 02

004    TOPIC:          SETS class XI                                                                     lesson No: 02

** We learnt that:
**SETS is all about COUNTING. Our duniya is the “size” that we choose for the problem at hand defining a criterion. Nothing else exists for us.

So, if it is only counting, then we should be able to ADD, SUBTRACT etc
Then two sets can be equal, unequal. And so on.

**We said we could either list out ALL the names of members placed inside the pudiya.
Or if we could formulate a simplified way of saying what we mean and what others can understand it makes our work easier and saves space and effort. Like all know what are real numbers, or integers, or natural numbers. Or there are 12 in a dozen. Or what is a prime numer, even/ odd numbers.

Like we could say
X: x is a vowel, size: we know there are only 5 vowels in English Language.
Or
X: x a real number, give limits of real number value.

**  A member, say, a student, can have many different characteristics on which we base our criterion, like:
Languages known to him
State to which he belongs
Games that he can play
His performance in % of marks etc

** BUT students of class XI (circle A) cannot also be at the same time be students of class X (circle B), We have given no criterion here.

The two classes have nothing in common. Call them DISJOINT.
EX1: STILL if we chose a criterion like games there will be students of class X and class XI who play kabaddi. So, if we were to form a team for our school (duniya) we might find that in class A of 40 students 6 play kabaddi; and in class B of 50, 11 play kabaddi.

Thus, what is common for this case? It is game kabaddi. There are 11+6 students belonging to two classes who play kabaddi.
Blue area means 6 + 11 =17 students from the two classes who play kabaddi.

Similarly, we can make several different combinations for including even class XII.
** Now big katora (UNIVERSE) has two katoras class X and class XI. Etc.
Set A and set B are both SUB-sets of the main set Universe.

Blue area is again a subset of setA AND setB. And so on.

**TOO UNDERSTAND: WHILE blue area is common to both what it means is this: Assume that color film of circle A is red and color of film B is Green, then if we place one above the other as shown then color of the OVERAPPING area will be blue. This means that circle A containing the common portion actually is = A complete + green portion from B that made it blue.
Similarly, B containing blue portion is actually= Blue full + green portion from A that made it blue.
Thus EX1 Above means is this: When two colored circles A abd B overlap, then A containing blue = 40 + 6 members and
B containing blue area= 50 + 11
But we know that these two classes had only 40+50 = 90 students. They cannot change.
Hence, we can say:
When A overlaps B it means: A + B – ( 5+6) This is called UNION of set A and set B. Read below also.

** And set of COMMON members is called INTERSECTION.

**A subset is a pudiya inside the bigger pudiya and formed by members of main set only.

**Note that we can make a pudiya only when we have paper to make pudiya. Thus, until we place things inside the paper, we do not have a set. Having made a bigger pudiya we must have paper inside this pudiya to make a smaller pudiya that has nothing inside it. A pudiya that has nothing inside it but needs to be available is called a NULL set.

**A set that is empty but it has to be available. It is written as {Ø}
Thus a bigger set A = { 1,2,4,5,6}  must have an EMPTY set inside it. That is NULL set is a sub set of ALL and ANY set.
**Thus set A is actually= {1,4,6,2,5, {Ø}}. NOTICE anything here? Students can sit anywhere in the class.
We don’t show this null set. Normally.

** now since we can count and the members have unique ID we can say
Set {1,2,3,4} = set {1,4,2,3}   Students can sit anywhere in the class.
Set {1,2,3,4,5} ≠ Set {1,2,3,4}
Set { 2,7,3,4,2} = set { 2,7,3,4} WHY?
Set { s,c,h,o,o,l} = set { s,h,c,l,o}
Set { Ram, Shyam, Sita, Gita, Ram} = set { Ram, Shyam, Sita, Gita } WHY?

** We can define Range OR INTERVAL for keeping our set a FINITE set as stated before. Like , 1<z<10  that is z has values NOT including 1 and 10.
OR
1 ≤ X≤ 10 meaning x has values 1 and 10 INCLUSUVE. Simple.

** Once we have learnt fundamentals then we can twist basic into more complicated looking terminology, like:
** POWER of SET: all possible sets that can be formed from given set including a NULL set. Thus, for a set {1,2} we have power =
{Null set} + {1} + {2} + {1,2} = 4 sets This 4 is called power of this set. TRY some examples. Why not {2,1} also?

** For better clarity we can show each pudiya as a circle inside a duniya. Then we can even show how much is common between subsets inside the Universe. Like

Here U is duniya, sum total of all subsets cannot be more than duniya.
A is a subset; B is a subset of duniya;  Blue portion is common to both A and B and thus is a sub-set of A and B, etc
So easy to look at and understand. This is called VENN diagrams.
See! A inside Universe; B inside A. Katore mein katora!! B is a subset of A. A remains a subset of U.
**A sub-set means a set formed from setA USING some members of A, thus it has to be INSIDE set A only.

** now we can define more terms like:
When some portion of A, B overlap. They both have something in common. Suppose 3 members from a of 20 members; and 5 from set B of 30 members can speak English, then total number of members speaking English is 3+5=8.

A and B unite to produce blue area. Only because they have something in common.
If there is nothing common in either A or B then Venn diagram will be:

We can now make any kind of diagram. We construct a diagram to describe our problem VISUALLY- easy to comprehend.
**We could also say setA and Set B INTERSECT in (3 + 5) members ie total number of members who can satisfy our criterion in both the sets.
** All sets put together cannot become BIGGER than Universe.

Now if something is not with me then it must be with someone else but still inside U. Therefore all that is NOT with me but when added to me becomes = U is called COMPLIMENT. It compliments me.
Like U = A + (U-A) here (u-A) is a complement of A.

**A little more:
**ϵ:  Means that member x belongs to set A
BϲA:  Means that set B belongs to set A ie B is inside A or all members of B are members of set A
** Remember we said that if we have a set A as {1,2,3,4} and set B as {3,5,6,7} then Union means that we ADD both the sets giving set bigger set as {1,2,3,4,3,5,6,7} But then each member must be UNIQUE in a set. Here in the combined set 3 comes twice and is not allowed. Therefore, Union means that ALL members of set a and all members of set B, BUT members common to both come ONCE only. But we know this.

**Similarly :
**Difference A – B reads subtract members of B from members of A. Naturally only members which are present in BOTH can be subtracted.

**INTERSECTION: members common to two sets.
** Imagine one set A as a colored transparent sheet of say Red color and B as a sheet Green. If we place one on the other we get a color blue. This portion blue is the overlap containing members which are in BOTH sets OR members of two sets which satisfy our criterion. Thus in these circumstances A = A + (items of B which are also present in A say n1)
And, similarly
B = B + ( items of A which are also present in B say n2)

**Now if we UNION A and B we have

A union B = A + B + n1 + n2 which is more than A + B; NOT possible. Therefore we must subtract (n1 +n2)
Thus A U B= A + B – (n1+n2) .

** What is (n1 + n2) : INTERSECTION. because both have these common members based on our criterion.
Then A U B = A + B – (intersection A and B)
You can use this understanding to make all formulas.

That is about all about sets.
Now read your NCERT book.

If you feel there is something lacking or you can add something to make this lesson better, please contact us at: easyfundamentals@rediffmail.com

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