Functions

The things missed in Mathematics!

1.       Again considering the same sets of horizontal and vertical lines.

2.       This time, the vertical lines are broken in between horizontal lines, making smaller vertical lines.

3.       Now the same intersections define a function from set of vertical lines to set of horizontal lines.

4.       What exactly changed when the vertical lines are broken?

5.       Now, for each vertical line there is only one horizontal line with which it intersects but, for any horizontal line there can be multiple vertical lines.

6.       There is also another interesting thing about this situation!

7.       If you remove any of the vertical lines from the scenario, it will still represent a function!

Few formalized representations:

Relations: A relation is any subset of A X B of two sets A and B.

Functions: A function f is defined from a set A to set B and represented by f:A->B

Relations

The things missed in Mathematics!

1.       You know many things about Sets and their operations and Cartesian Product.

2.       So, putting things in a simple and straight manner.

3.       There are two sets, one of horizontal lines and other of vertical lines.

4.       A Relation is just the number of places they intersect with each other! (refer the picture below).

5.       To expand this definition, you can assume these lines as being anything!

6.       Please post your assumption as what you assume for horizontal and vertical lines, and I will reply with how they are intersecting!

Cartesian Product of Sets

The things missed in Mathematics!

1.       I have very good neighbours.

2.       Each one in our family knows each one in our neighbours’ family.

3.       That is to say, each one in our family is related to each one in our neighbours’ family.

4.       The name of this relationship is, simply knowing each other!

5.       Hence, our relationships form the Cartesian Product between our family and our neighbour’s family.

6.       Even ZeuS tells us the rule of Cartesian Product.

7.       Refer the diagram. Consider two areas on a plane namely Z and S.

8.       Z has points A and B, S has points C and D. These two areas are also sets (sets of points!).

9.       If we overlap Z and S, the symbol as shown in the figure is formed with A,B,C and D as edge points.

10.   Look at how A is connected to C and D, B is connected to C and D and similarly C and D also connected to A and B.

11.   And this symbol also has the cross (X) in it to indicate that it is Cartesian Product.

Difference of Sets

1.       There is one more thing that I didn’t tell about what I like to wear on my date.

2.       Well, I will certainly not wear any formals on the date. So I will remove all the formals from my choices.

3.       I have a choice of mix of formals and casuals and mix of formals and jeans wear.

4.       I will remove all the formals from the above two choices, and will take only those things that will remain.

5.       Removing formals, I have few things from my casuals and few things from my jeans wear. (I don’t have everything, as I have only considered Intersections and on the Intersections I am now applying my new rule)

6.       This rule of removal is basically what is called Difference of Sets.

Few formalized representations:

Difference between two sets A and B is denoted by A – B and is defined by: All elements of A that are not elements of B.

Intersection of Sets

1.       Ok. Having the choices to dress up. To decide further I will add one constraint now.

2.       I will consider only the combinations from different types of dresses (that is mixes only) that will fit with each other.

3.       That means either a shirt from casuals that look good with a trouser from formals.

4.       That means those combinations that have features common to each other but usually belong to different type of dresses.

5.       Considering that I will have the following combinations to try:

6.       Mix of formals and casuals.

7.       Mix of formals and jeans wear.

8.       Mix of casuals and jeans wear.

9.       So I have another set of dresses to consider that will have common features with each other.

10.   These are the dresses that belong to the grey area between two different types of dresses.

11.   The thought that I have followed here is what we call Intersection of Sets.

Few formalized representations:

Intersection of two sets A and B is represented by A ∩ B and is defined by: Elements that are common to both A and B.

Union of Sets

Concept: Union of Sets

Storyline:

1.       I was looking at my wardrobe. (Because I am still behind that girl and want to look impressive).

2.       I have various sets of attire – formals, casuals, jeans wear. (These all are sets)

3.       Now I want to try various combinations from all of the types of dresses I have.

4.       Guess what will help me to form combinations and try them? Set operations!

5.       First thing I will try is by using Union of Sets. So I will have following options to consider by following the Union operation:

6.       Either wear formals or casuals or mix of both.

7.       Either wear formals or jeans wear or mix of both.

8.       Either wear casuals or jeans wear or mix of both.

9.       So now I have three new sets of attire to keep under consideration by following Union of Sets!

Few formalized representations:

Union of two sets A and B is denoted by A U B and is defined by: Elements of either A only or B only or both.

Operations of Sets

1.       You are very happy with your family, and your set of your family members is complete with all the family members in it.

2.       But, soon you will be getting married to the girl/boy you loved.

3.       So you have a problem now, you have your set of family members that has this rule that it will only contain your own family members.

4.       Now how you will allow your spouse and his/her family members to come under one set?

5.       You basically have to change the meaning of your family. To do this you need the ways to change this meaning and that means changing the rule by which the set is made.

6.       The ways in which you can bend the rules that make a set is by performing Operations of Sets

Few formalised representations:

There are three basic Operations of Sets, namely:

1.        Union

2.        Intersection, and

3.        Difference of Sets

Complement of a Set

1.       Yesterday night I went to the prom night. (That’s why you can’t see any new post yesterday night from meJ).

2.        This prom was the whole Universe for the night for me. (Consider it to be my Universal set)

3.       I saw a girl, so magnificently beautiful and gorgeous. I just looked at this girl, forgetting about the rest of the crowd.

4.       I made a set that consists of only me and that girl, together! J

5.       Rest of the crowd in the prom was the Complement of our Set that consisted both of us!

6.       Then suddenly, someone from my friends calls me, and then I came to my senses and was again aware of this compliment of our set that was already there! J

7.       And I am still in the pursuit of actually materializing the set that I just had in my thoughts from that point of time…. J

Few formalized representations:

Complement of a Set A is represented by A’ (A primed) and is defined as everything from Universal set U except the things in set A. And A is a subset of Universal set U.

Universal Set

1.       Suppose, while doing your all cleaning there was an ant on one of the pieces of paper kept near C.

2.       And now, because of you she is inside one of those small bags inside the big bag P.

3.       She comes out of her small bags and observes the things around her. Now whatever she observes, it is the only things that she is going to live with in rest of her life.

4.       That big bag P is now her entire Universe and for her, this bag P is the Universal Set.

5.       P is Universal Set for our poor ant because whatever she can find, it all belongs to the set P.

6.       Pity on the poor ant! Can’t you be more careful while doing your work! J

Power Set

1.       Enough of breaking and spreading pieces of paper. Time has come to now do cleaning!

2.       After breaking sets and tearing papers, how about putting them together now?

3.       So, by now you have got your paper pieces gathered near B, C, D, E, F, etc…

4.       Find small paper bags and put the pieces near each of the alphabets in separate bags.

5.       Put all these bags in a big bag and write P on the big bag. Also put an empty small bag inside the big bag.

6.       This P bag, my friends is your Power Set. Now you can happily go and dump this P bag and get rid of all your garbage from your table! J

Formalized representation:

Power set of A is P.

P = {ϕ,B,C,D,E,F…}

How to Make your own subset?

1.       Take a piece of paper, write a very big A on it covering the entire space.

2.       Tear this paper into at least 10 pieces.

3.       Write B and C on the table near two diagonally opposite corners.

4.       Divide the pieces of the paper in two parts; collect one near B and another near C.

5.       That’s it! You have your two subsets B and C of A.

6.       What? You expected more steps? Ok, then you can go ahead and can even divide and tear pieces near B and C further and make more collections of pieces near the markings D, E, F,etc…

7.       But remember, finally you have to clean your table!

Few formalized representations (this means the serious stuff, if you have not got it yet):

B and C are subsets of A is represented by B C A, C C A

Φ is subset of all the sets

Subsets

1.       It happens with majority of children when they play, they are so curious to know about their toys that they start opening its screws, or break it and still break more to know more about their toys (but they may end up in dismay sometimes!).

2.       Mathematicians are no different (they are crazier than those children). They want to break everything infinitely! (And as with the children so is with Mathematicians, they may end up in dismay sometimes!J).

3.       So they started to break Sets in as many possible ways as they can. And these pieces they called as Subsets.

4.       Subsets are also sets that are made from a set by splitting it into different sets that contains few of the selected things from the main set.

5.       Well, we are dealing with mathematicians here, so we have to take extreme meanings also in consideration.

6.       Few here really means two extremes of few are possible – one extreme is the same number of elements in the main set and next extreme is no elements at all.

7.       Simply put, an entire Pizza is a set and all of its pieces are its subsets.

8.       Even the entire Pizza is also its own subset, and when you have eaten all of the Pizza and no pieces are there, that is also a subset of the entire Pizza!

9.       So, next time when you don’t get a chance to bite on the Pizza among your friends, don’t worry you still have got its one of the subsets! J (I warned before also, mathematicians take a pride in making you crazy like them)

Few formalized representations:

Φ or {} – empty set

{Some rules or things listed comma separated} – Representation of a set

Finite and Infinite Sets

1.       So, we understand what a set is. It contains something, bounded by some rules.

2.       Let us talk about the nature of the rules that define a set.

3.       First thing that we have till now noticed obviously (I am not sure how much obvious it is!) is that these rules must help us to decide with certainty whether something will belong to the set or not.

4.       Next thing about the rules is that how many things they allow in the set.

5.       If they allow a number of things that you can count or in advance know how many things will belong to the set, then it is a finite set.

6.       If the rules allow a number that you can’t count or you know that there are always more, as you continuously can find more and more things to be put in the set, it is then an infinite set.

7.       That much said it is really difficult to swallow the fact that how can someone make a rule that he/she may not know exactly how many things to put in the set?

8.       Well, if you think you can only think of the rules that will give you some finite set then you are in your senses and have not gone crazy! J

9.       But, take the example of a crazy person who wants to make a set with the sand particles from a sea beach!

10.   He clearly knows what to put in this set, and he is also sure he is going to find many things to put in his set. The difficulty is he can’t ever tell in his lifetime that how many sand particles will finally be there in his set!

11.   So there can be some crazy rules possible, and to allow for this craziness, Mathematicians (who are equally crazy and take a certain happiness in making others crazy too J) created this concept of infinite set!

12.       So, clear about finite and infinite set. Now, where should the empty set be classified, finite or infinite?

13.       What you know about empty set is that it has nothing, you are certain about that it contains zero elements. That is you know how many things it has, it has zero things! That’s why it is also a finite set.

The Empty Set

1.       How something with nothing in it could have a value to us?

2.       Simply put, a set that contains nothing is an empty set.

3.       What a useless concept, isn’t it?

4.       Well not really. There are many things around us, which we dislike to have with us.

5.       But we are also very shy to admit that we don’t have anything that we like.

6.       So we simply choose to say in a different way. We say that, well guys we have a set that contains everything that we like to have, but only thing is that it contains nothing as we could not find anything that we liked.

7.       Empty set is this gracious way of saying that we don’t have anything, mathematically! J

8.       There could be some rules made by us, for which there will be nothing to come in our set. So, to save our hesitation to bluntly say that we don’t have anything, we will have Empty set.

Origin of Numbers

1. As I said, that I can't mathematically prove whether a person has knowledge about numbers or not, so well it will be better if I can ensure that knowledge is re-iterated here.
2. Well most of us can answer this question pretty easily, that numbers came into existence to get us the counts of "things".
3. Numbers don't only have this responsibility to get us the counts, they also provide us with a value to work on.
4. And even deep is the meaning of the "things" that are counted using the numbers. (Yes, its not the money you are counting using these numbers).
5. Let us take a story from the city of Mathema* to understand numbers.
6. All people in Mathema are born blind, they can't see anything. They have learnt to recognize the following shapes, by touching:
     0   1   2   3   4   5   6   7   8   9
7. Everyone in Mathema knows to make these shapes, and only these shapes. As all the generations in Mathema were blind, they could not communicate how many or how much they possess to anyone else by physically showing them the objects.
8. They only have the means of touching to communicate. They slowly learnt to associate each of the shapes above to the number of fingers.
9. 0 - no fingers
    1 - one finger
    2 - two fingers
    3 - three fingers
    4 - four fingers
    5 - five fingers
    6 - six fingers
    7 - seven fingers
    9 - nine fingers
10. The same association was spread to entire city slowly, communicating by touching the shapes and the number of fingers to be associated with that shape.
11. Now coming back to our world from Mathema, the shapes that were recognized by the Mathema people are just symbols, they don't have any meaning unless we provide them some meaning.
12. The meaning to these symbols were given by associating them with the number of fingers by Mathema people. This is the meaning of that symbol, and this is what we call the value of that symbol.
13. We are so used to the meaning of these symbols, that we take the shapes same as the value, they are different things.
14. So, now you must agree that we are not actually "counting" the money we have! We are counting the fingers! :)


* Mathema is the root for the word Mathematics. The city of Mathema is a creation of imagination and there is no such city by this name in past or present. If there happens to be a city with the same name, then it is purely coincidental.

Set Definition

1.       So, sets are really simple to explain and people to understand. But do we really understand what sets are?

2.       In simple words, sets are collection of anything defined by rules such that the rules can clearly tell us whether to put a thing to that set or not.

3.       But what this “anything” is? Basically, what is the “thing” that we are talking about? Are they objects? Why such a concept like set has come into existence? How is it helping us in Mathematics or in any other part of our life?

4.       Let us consider one example; your family is a set. How? It consists of your family members (a rule) and at any point of time you or any of your friends or family can, without ambiguity, decide who is your family member is and who is not.

5.       Even for defining a set, we need to have some particular knowledge about the things that makes the set.

6.       If you give the above definition of the set to a total stranger. Well, that person can’t decide who your family member is or who is not, because he doesn’t know about your family.

7.       So for that person, it is not a set. Once he is provided with the knowledge about your family, then only he can realize that this is a set.

8.       Hence, in Mathematics also when we are defining a set, we should know about the numbers. This knowledge is assumed to be possessed by everyone (but we can’t prove this mathematically ;))

9.       Now coming to the decision about classifying something to a set. The rule says members of your family, so by the knowledge of what makes a family member is basically checking if that member is in relation to you. The relation can be sister, brother, mother or father.

10.   The rule is basically, is a collection of relationships that can be possible between you and any of your family members.

11.   Now coming to what “things” can make a set, it will amaze you that the point 10 above is also a set! J

12.   Well, you are getting to know about Sets!