class XI Relations and Functions

class: XI      sub: Maths          chap: 2   Relations and Functions            lesson: 01

If we told you that boy is Ram. You are bound to question, “so what”?
If we collected (Ram, Shyam, Ravi ) as a set again you might raise the same question. But if we also told you “they are all my friends. Now it makes some sense. Why? Because we told you their RELATION with me. May be now you can trust them.
Similarly if we said, “that girl is Rekha”- it means nothing. But if we told you she is sister of Ram, Shyam and Ravi, it makes sense. Because they are all brothers and a sister.
We have CREATED a relationship between three boys standing on one side as a group and one girl standing on other side.

** We can call boys= ( R,S,Ravi) WHY?
And girl = (Rekha) WHY?

** We all have drawn a graph. It has X-axis and a Y-axis. Suppose we asked you to write the following:
(2,1), (2,3), (2,5), (2,7) and plot them. You quickly tell us: this is a straight line parallel to Y-axis at a distance of 2 units from origin. Also, that the numbers we gave you are CARTESIAN CO- ORDINATES.

Similarly we could draw lines like 3,5, 4,5, 7,5 giving us…. WHAT?

** Suppose we made two sets as:
{2}                          and                       set {1,3,5,7}
Suppose we held digit 2 from left set in our hand and made sub-groups taking one digit at a time and RELATED them as
2,1          2,3          2,5 etc and formed a new set in which the members are having two elements in place of one, this set looks like
{(2,1), (2,3), (2,5), (2,7)}
This we immediately recognize as co-ordinates in TWO dimension system- CARTESIAN and let us call this new set made as above a CARTESIAN PRODUCT of two sets above. If there had been more than one number in first set as (2,6) we would do same OPERATION using 6 like we did with 2 and made the PRODUCT set bigger as follows:
{(2,1), (2,3), (2,5), (2,7), (6,1), (6,3), (6,5), (6,7)} and so on.
** We note that individually 2 or 6 means nothing. But when we RELATE it with members of another set and define a relation then a MEANING occurs.
** CARTESIAN PRODUCT: if we have two sets A and B then Cartesian product can be obtained by holding one element from set A and RELATING to EACH item of set B in SEQUENCE from left to right till we reach the end of B; then repeat the same taking second element from A . NOTE ALL elements are taken in ORDER. You cannot pick and choose.
** ALL elements of FIRST set taken together is called DOMAIN. It is like all properties that we want to use. Naturally we should choose only what we require. If we want to use 5 elements then we CANNOT choose a domain of 6 elements.
**ALL the items of second set taken together is called CO-DOMAIN.
**Number of elements from set B second set to which we ACTUALLY apply is called RANGE of A. Thus Range can be less than or equal to co-domain.
**What have we understood:
One boy can be brother to many sisters.
If there are standing 4 girls, this boy may be brother to only 2. Define range, domain, codomain here.
In graph example there can be many corresponding values of Y for same X. This means ONE to MANY relationship.Every action happens in ORDER, in sequence. You cannot break this ORDER.
We took example of two sets. We can take THREE or more sets also. Procedure remains the same. Say set P,Q,R.
Then Cartesian product will be :
Do this on Q and R giving set say K; then do same thing using P and K. IN ORDER.

**We know that a set can be written in roster form and set builder form. In roster form we need to write each element/ subset. Why not state limits here?

**In set Builder form we state a relation or a formula or a fact that all understand. And we also state the limits or Domain. WHY?

**The above is okay for data which has names of students, names of teachers, names of schools etc. Everything must be in  ……… form. Suppose we were to say:
Ram, a student of class X, of DAV school, a monitor in class, son of Laxman, resident of Mumbai etc.  Only roster method shall be used.

**FUNCTIONS: we could have said a parallel line to Y-axis at a distance of +2.
Or plot of a line Y = mx +c
Or y = X2
meaning that there exists a RELATION between Y and X which we can represent in the form of a formula that all understand.

**It is like saying: Put an apple into the cutting machine and out comes 4 pieces of apple.

Or for y = X2,  we put a value 2 for x variable into some machine and machine gives output Y as = 4 and so on. This machine is called FUNCTION. This is also a relation.
We could then write output as a set like:
{(2,4),)3,9),…(7,49),…} in Roster form.
OR Say {(x,y)|y=x2 , for all real numbers OR give a domain and range} in set builder form.
Suppose we take +2 and -2. Square of both is = +4. MEANING MANY to ONE relation.
Or Y = 5x a sloping line. Here for each different value of x there is only one value of y. MEANING ONE to ONE.

Definition:  function f(x)| x → Y ; means  Y = f(x)
That is x gives result = Y when passed through a machine called f(x)

That is Cartesian product using a function as relation gives ONLY ONE result. Even if there are many elements in the domain that also give the same result.

LOOK at the following figs and speak out what you understand.

NOT a function. WHY

Now read your book.