Some formula and questions tips for qualifier.( particularly for non maths back ground)

[Malabika Guha Mustafi]

We may expect some of this area from where questions can be asked-

1. Set

Please  clear your basics about set, power set, cardinality and set comprehension.Many of us made mistake in Mock overlooking the set cmprehension .T={n|f(n) is prime }

Here, n should be  considered as  elements not f(n). 

2. Relation ( I will post the function and relation notes here ).

3. function

Altogether 5 questions came from this topics in Mock.

2. Section formula  related questions

If we can remember the formula then the computation is not very difficult.

[Same for Y also , (Y-Y1)/Y2-Y]

3. Problem related parabolic equation.

Whenever it is mentioned maximum or minimum, It is actually related to the  x Value at Vertex  -b/2a. and corresponding Y value{( -b^2/4a) +c}

4. Problem related straight line (slope point form. Y=mx+c)

First compute m ( from difference of Y value /difference of X value)

Then proceed.

m= rise/run = change 

5.  Problem related reflection of a line along X axis or Y axis

If reflection is done along X axis then Y coordinate only change the sign.

If reflection is done along Y axis then x  coordinate only change the sign.

6.  Problem related to Parabolic equation where maximum or minimum is not required but asking for other thing.

 Draw the parabola first and try to solve it ( it may be root related problem. It means, If we put Y=0 fro ax^2 +bx +c type equation and  X=0 for aX62 + bx + c type of equation.The quadratic solutions are known as roots.

root means , pictorically  where the parabola cuts X axis ( it may be in  two places, one place or not at all for ax^2 + bx +c equation)

root means , pictorically  where the parabola cuts Y axis ( it may be in  two places, one place or not at all for ay^2 + by +c equation)

7. Please remember the distance formula, Two point line equations, all line forms.

The questions are quite easy to compare other problems and direct.

8.The problem related parallel line and perpendicular line.

Remember two m produce

m1 X m2 =-1 [ for perpendicular]

and m1=m2{ for parallel line)

 Please Keep your logic and nerve steady. It is easier to say , I know. 

All the best .

[Malabika Guha Mustafi]

Relation Types

A) Reflexive 

If (a,b ) is in  reflexive relation then ALL elements  (a,b where a=b) should be held the condition as true or all elements to be present.

Examples :

1) A relation R is on set A (set of all integers) is defined by “x R y if and only if 2x + 3y is divisible by 5”, for all x, y ∈ A. Check if R is a reflexive relation on A.

x is an element of set A

y is an element of set B

when x=y

then 2x+3y=5x=5y so divisible by 5

The relation is reflexive.( it should be true for all values).

2)A relation R is defined on the set of all real numbers N by ‘a R b’ if and only if |a-b| ≤ b, for a, b ∈ N.

Now when a=b

then |0| not equal to or less than b.

So, the relation is not reflexive.

3)A relation R on the set A by “x R y if x – y is divisible by 5” for x, y ∈ A. Check if R is a reflexive relation on set A.

when x=y

then x-y=0 which is divisible by 5 for all values of x,y so it is reflexive.

4)

Suppose the relation 

R={(Snail, Frog), (Bird, Bird), (Fox, Frog), (Snail, Fox)} is defined on the set S={Snail, Fox, Bird, Frog}.

Here, (snail,snail), (Fox,Fox) , ( Frog, frog) elements are missing so the relation is not reflexive.

If x(a,b) and y(c,d) then the relation for x X y  will be Reflexive only if all the diagonal elements of the cartesian product(a,a),(b,b),(c,c) and (d,d) should be present in the set.

B) Symmetry

If R is symmetric relation, then R  =  {(a, b), (b, a) / for all a, b ∈ A}

That is, if "a" is related to "b", then "b" has to be related to "a" for all "a" and "b" belonging to A. 

Examples:

1) Let a, b ∈ A.  

If "a" is brother of "b", then "b" has to be brother of "a".

R  =  {(a, b), (b, a)}

So, R is symmetric. 

2) Let A  =  {1, 2, 3} and R be a relation defined on set A as

R  = {(1, 1), (2, 2), (3, 3), (1, 2)}

R is not symmetric since (2,1) is missing. (Though it is reflexive ).

3) Let A  =  {1, 2, 3} and R be a relation defined on set A as

R  = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (2, 3), (3, 2)}

 R is in symmetry relation. (It is reflexive also).

4) Suppose the relation 

R={(Snail, Frog), (Bird, Bird), (Fox, Frog), (Snail, Fox)} is defined on the set S={Snail, Fox, Bird, Frog}.

Here ( Frog, Snail ), ( Frog , Fox) , (Fox, Snail)are not present so R is not symmetry.

C) Transitive

Let us consider the set A as given below.

A  =  {a, b, c}

Let R be a transitive relation defined on the set A. 

Then, R  =  { (a, b), (b, c), (a, c)}

That is,

If "a" is related to "b" and "b" is related to "c", then "a" has to be related to "c".  

Important Note :

For a particular ordered pair in R, if we have (a, b) and we don't have (b, c), then we don't have to check transitive for that ordered pair. 

So, we have to check transitively, only if we find both (a, b) and (b, c) in R. 

Examples:

1)Let A  =  {1, 2, 3} and R be a relation defined on set A as

R  = {(1, 1), (2, 2), (3, 3), (1, 2)}

 The relation is transitive.

Note : For the two ordered pairs (2, 2) and (3, 3), we don't find the pair (b, c). So, we don't have to check the condition for those ordered pairs.  

2)Let A  =  {1, 2, 3} and R be a relation defined on set A as

R  = {(1, 1), (2, 2), (1, 2), (2, 1)}

so the relation is transitive.

3) Let A  =  {1, 2, 3} and R be a relation defined on set A as

R  = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (2, 3), (3, 2)}

In the table above, for the ordered pair (1, 2), we have both (a, b) and (b, c). But, we don't find (a, c). 

So, the relation is not transitive.

4)Suppose the relation 

R={(Snail, Frog), (Bird, Bird), (Fox, Frog), (Snail, Fox)} is defined on the set S={Snail, Fox, Bird, Frog}.

(Snail,Fox) and (Snail,Frog) leads to (Fox,Frog) 

Another pair is not available.

So the relation is transitive. 

**Important things to remember, for transitivity verification no need to have the presence of all pairs like reflexive or symmetry but if a pair is present then it should be held the condition as true.

D) Asymmetric 

 A relation R on a set A is called asymmetric if no (y,x) ∈ R when (x,y) ∈ R. Or we can say, the relation R on a set A is asymmetric if and only if, (x,y)∈R⟹(y,x)∉R.

Example

1)

 If A = {2,3} and relation R on set A is (2, 3) ∈ R

Here, (3,2) is not a part of set[ it is asymmetric] [It is not reflexive, not symmetric and transitive]

**It is opposite to symmetry so Asymmetric relation can not be symmetric and reflexive.

E) Anti Symmetry

 The relation R is said to be antisymmetric on a set A, if xRy and yRx hold when x = y. Or it can be defined as, relation R is antisymmetric if either (x,y)∉R or (y,x)∉R whenever x ≠ y.

since it takes x=y so, it can be symmetric and reflexive.

Example

Let R be a relation on a collection of sets defined as follows, 
R={(A,B)|A⊆B} 

here , the B is subset of A is only true when A=B so it is antisymmetric.

F) Equivalence

when the relation is reflexive and symmetric and transitive.

Examples

1)The relation “is equal to”, denoted “=”, is an equivalence relation on the set of real numbers since for any x, y, z ∈ R:

since x=y means it is reflexive

y=x so it is symmetric

x=y, y=z so x=z thus it is transitive.

2) The relation “is less than or equal to”, denoted “≤”, is NOT an equivalence relation on the set of real numbers. For any x, y, z ∈ R, “≤” is reflexive and transitive but NOT necessarily symmetric.

[Malabika Guha Mustafi]

Function Types. 

Mainly three types need to remember.

1.one to one ( Injective)

When for  each x value there is a distinct y value.[y is the function of x]

If two different x value generate same y value then the function is not injective. ( think about injection, no two persons are allowed for one injection)

2. Onto ( surjective)

Simple approach, just remember your range ( the value you will get from the function directly) should be same as co domain.( the domain of the y which is  function of x).

If your codomain defined as real number but the range is taking only integer or rational or natural number, then the function is not surjective  or onto. If your codomain defined as range of natural number but your range is not covered full range just picking few elements then also the function is not surjective.

3. Bijective

By definition , the  bijective function is injective and surjective[ both onto and one to one].

[Harikumar P]

Thank you Malabika Guha Mustafi.

Your Notes and tips are very clear and helpful 

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[Pragati Mungekar]

Thanks Malabika.

Very helpful tips .

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[Puneet Bhagat]

Thanks Malabika. The notes are very helpful.

All the best. Stay blessed.

[Sanchi Priyanshu]

malabika how much marks u  got in maths and computatioal mock ,i  just want to eagerly know,by the amount of knowledge u  have?

[Malabika Guha Mustafi]

I am just a learner.

 I could not submit my Maths Paper.

[Anand Iyer]

I always wonder what our tutors might be thinking after reading our jottings around here:)  They might be either lol or col...maybe rofl or rofc. 

NOTE: I just coined a few terms here:)

[Malabika Guha Mustafi]

Lol definitely :)

[SAKSHI GUPTA RAMTEKE]

Thanks Malabika.

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[Malabika Guha Mustafi]

Please make the correction( Draw the parabola first and try to solve it ( it may be root related problem. It means, If we put y=0 fro ax^2 +bx +c type equation and  x=0 for aX62 + bx + c type of equation.)

It should be ay^2 +by +c.

Thanks

[Rajashree Govindan]

Thanks Malabika for these extremely useful tips.

Regards

Rajashree

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[Tejasvi Hegde]

This is great pointer @Malabika

[Malabika Guha Mustafi]

Thank you everyone. 

[Akshay Malik]

For people finding it difficult to solve the maximum-parabola problems

[Akshay Malik]

When trying to find whether a function is injective or surjective , using the arrow-diagram makes things very clear.

[Akshay Malik]

How to recognize a non-injective function.

I can illustrate more if anyone has any doubts.

[nikhil kambli]

thanks @malabika @akshay

here are my hints

formatting may be not good 

(i couldn't submit the math scored pathetically in CT focusing on stats and did really bad on stats )

thank you for the tips.

(i am having trouble with surjective injective and bijective and functions )

now if i have understood correctly every element of x should have single corresponding element in Y  injective

and for surjective every element of y should have a pre image in x

so can a function be surjective and not injective?

could you give an example

__________

for function to be reflexive we have to check if the elements of x=y are present in that set and all should be present.

for function to be symmetry we have to check if x=y and all should be present in the set 

and for function to be transitive we have to check whether a single element is present satisfying the condition  a=b b=c whether a=c but all elements neednot satisfy the condtion 

unlike the above 2

___________

for a function x-->y

domain is all numbers of x

co domain  is all numbers of y

range is numbers of y which can be achieved 

so range is subset of co- domain

_____________________

number of subsets in a powerset is 2^n where n is number of elements

_______________________________

rectangular co-ordinate system

distance from a point is  sqrt(( x2-xq)^2 + (y2-y1)^2)

section formula   m/n = x-x1/x2-x =y-y1/y2-y

x = mx2+nx1/m+n

y= my2+ny1/m+n

area of triangle

= 1/2 | x1(y2-y3) +x2(y3-y1) + x3(y1-y2)|

something more.

When the length of three sides of the triangle are given, the area of a triangle can be found using the Heron’s formula.

Therefore, the area of the triangle is calculated using the equation,

A = sqrt( s(s − a) (s − b) (s − c))

Where a, b, c are the side lengths of the triangle and s is the semi perimeter

The value of s is found using the formula

s = a + b + c2

area of trapezium 

is  = 1/2 ( sum of parallel sides) * distance between them

if parallel to x axis y co-ordiate of both points should be tkne

if paralle to y axis x co-ordiate of both points should be taken

__________

slope = rise/run = tan theta  where theta is the inclination of the line with positive x axis measured in anticlockwise direction

also remember m = tan(180-theta) = -tan(theta) = (y1-y2)/(x1-x2)

parallel lines m1=m2

perpendicular lines  m2*m1 = -1

relation between two lines and their slope

= (m1-m2)/(1+m1.m2)

two point form  (y-y1)/(x-x1)= (y2-y1)/(x2-x1)

non vertical lines are perpendicular if an only  if m1*m2 =-1 

equation of line using point slope form  (y-y0)=m( x-x0)

y intercept  y=mx+C

x intercept x=m(x-d)

horizontal line y =a parallel to x axis

vertical line x=b parallel to y axis

intercept form

x/a+ y/b =1

____________________________________________________

general equation

Ax + By + C

m= - A/B

y0-x0= -C/B

y = mx + c ( y interecept)

y = m(x-d) ( x interecept)

c= -C/B

 d= -C/A 

___________

b= -C/B

a= -C/A

____________________________

equation of parallel and perpendiuclar lines in general form

ax1+ by1 + c1= 0

ax2 + by2 +c2 =0

for aand by not = 0

parallel linesif p

 a2b1= b2a1

perpendicular lines a1a2 + b1b2 =0

_________________________________________-

distance of a point  =  (|Ax1 + By1 +C |)/ sqrt( A2+b2)

distance of parallel lines = (|c1-c2|) / sqrt (a2+b2)

____________________________

parabola = Ax2 + bx+ C

slope of parabola at a given point  y= 2ax+ b 

x intercept = -b/2a

( where parabola is maximum or minimum)

__________________________________________

i hope the list is comprehensive may not be correct 

please do correct and append..

thanks to @m

f

[nikhil kambli]

in heron formula s =( a+ b+ c)/2  there has been formatting error

[nikhil kambli]

sideways parabola is not a function 

slope of line perpendicular to x axis or parallel to y axis is not defined

every function is a relation BUT every relation is not a function

++++++++++++++++

[Tejasvi Hegde]

Its great reference.

Just a silly question...

How do you all remember Formulas ? :)

Knowing ways everyone do may help.

I keep writing them while solving and also try to recall when not doing anything. 

[Malabika Guha Mustafi]

Aways try to map the diagram associated with formula. Mapped  mentally with diagram other wise formula is hard to remember.

Example

For sectional formula which one is most cumbersome .

Think about a line one end(x1,y1) and other end(x2,y2)

(x, y ) will be in-between.

Fist part is m and second part is n.

now, we need to criss cross multiplication.

n to (x1,y1) and m to (x2 ,Y2)

@ Tejasvi and others who are struggling to remember formula , it may help you. 

Vertex formula should come with your parabola image not alone.

For line equation , just emphasis one formula and other formulas can  be derived from one.

All pictures are different so brain can recall precisely but if you try to remember only formulas which are only  combination of some alphabets and numbers then it may become messy during recalling.

[Tejasvi Hegde]

That's fantastic explanation @Malabika

Thanks

[Debajyoti Biswas]

Just to answer the doubts of Malabika and Anand, we every day look forward to all of your nice answers and discussions, note who is more active on forums, and discuss among us and the admin and support team on how to update the course more to help all of you in learning. Some of the names we are familiar with, as good students, we are identifying more students as well. 

[Malabika Guha Mustafi]

Thank you very much for your encouragement . Team put an enormous efforts and I really like the journey . Thank you again all the team member.

[ar Memon]

THANKS A LOT ..........I HAVE.NO WORDS FOR THANKS AS THE WAY YOU HELPING OTHERS ,,,,,,,,,,,,,,,,

[nikhil kambli]

Use of visual while remembering formula

Very interesting

On Thu, 19 Nov 2020, 13:40 Malabika Guha Mustafi, <malabi...@gmail.com> wrote:

Always try to map the diagram associated with formula. Mapped  mentally with diagram other wise formula is hard to remember.

Example

For sectional formula which one is most cumbersome .

Think about a line one end(x1,y1) and other end(x2,y2)

(x, y ) will be in-between.

Fist part is m and second part is n.

now, we need to criss cross multiplication.

n to (x1,y1) and m to (x2 ,Y2)

@ Tejasvi and others who are struggling to remember formula , it may help you. 

Vertex formula should come with your parabola image not alone.

For line equation , just emphasis one formula and other formulas are can derived from one.

All pictures are different so brain can recall precisely but if you try to remember only formula which is only some alphabet and numbers then it may become messy during recalling.

On Thursday, 19 November 2020 at 13:31:36 UTC+5:30 Tejasvi Hegde wrote:

[Malabika Guha Mustafi]

Effective tips for computation of triangle area:

If the coordinates of vertices  are given ,  try to draw it first. Maximum time , we don't need to put the big formula. For example 3.4 q.5

you can easily compute by 1/2 x base and height.

[Dhannya POD]

😁😁

On behalf of the team, I would say that the whole support team is truly happy and humbled to see such selfless contributions from many of you. 

Keep posting.. I am sure there are many learners who are silently thanking the contributors!

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IIT Madras.

[Tejasvi Hegde]

Dear Ms. Dhanya ,

Heartily thanks to you and your team.

You and your team held hands of all of us and tried to prepare us for the best, I really appreciate that tremendous effort of you all.

It has been wonderful journey with good company of fellow learners, the names top of mind, @Anand  and @Malabika 

If in case, for some the reasons if I can't make to next stage, It is going to be a experience to remember.

And personally,  I want to thank you for tolerating my criticisms and comments. 

Thanks again!

and Good Luck to all friends!

[Malabika Guha Mustafi]

Thank you all. I am really thankful to Team, all the contributors  who answered my enquires and clear other's doubts . I am not taking any name because  if I miss  a name it will be unfair. I am thankful to all of you who personally message me and wish for my exam. I am specially thankful to all who asks questions . Your doubts push me to think the subjects more deeply. 

Last but not least, special thanks to all the people who make excellent notes,shortcut formula chart, giving book reference and  site reference..All the best. 

[nikhil kambli]

thanks to everyone. last few days i have been working and studying so this forum and the tips have been a lot of help

i hope the best for all of you . and hope we all get to discuss much more further.

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[Ganesh K Pillai]

That was a really a  'Lot of  HELP' Malabika  ...  Thank You so very much  !!! 

I just hope I remember all these ... actually I am not used to remember anything at this point in my life ...  

Now .I am . Learning to remember /memorize ... 

Team IITM => Do I really need to memorize / remember formulas   ;-)  ... is it a requirement  to solve the Qualifier Exam .

 I would not have asked this if I was in School Though   ;-) ...,  

[Ganesh K Pillai]

 

Dear Malabika / Team IITM -

Please Refer attachment -

Does The element (Bird, Bird)  contributes in any form to transitivity ? 

The explanation says  ' similarly ( Bird, Bird)  in R ' , Hence R is Transitive ... So I m confused ...   

Thanks 

[Malabika Guha Mustafi]

Dear Ganesh,

Trying to clear your doubts.

For checking reflexive relation, please keep in mind that all diagonal elements of cartesian products should be present. I means for the above examples , (Snail,snail);(bird, bird); (Fox,Fox)and (frog, frog)  are the diagonal elements of this set. But only (bird, bird) is present but no others elements so , it is not reflexive.

For transitivity, we need to prove  for only those elements are present . If (a,b) and (b,c) are  there then(a,c) should be present for transitivity proof otherwise the function is not transitive,
But if only (a,b) is there and no other external condition is stated for the set then without any presence of (b,c)and (a,c), we can tell that the relation is transitive.

Note : For the two ordered pairs (2, 2) and (3, 3), we don't find the pair (b, c). So, we don't have to check the condition for those ordered pairs. 

Important Note :

For a particular ordered pair in R, if we have (a, b) and we don't have (b, c), then we don't have to check transitive for that ordered pair. 

So, we have to check transitively, only if we find both (a, b) and (b, c) in R. 

Here , (bird, bird) can be compared with the above mentioned (2,2).

Actually, both the logic [ 1)when (a,b) and (b, c) are there we need to check (arc) should be there like (snail, frog) and (fox, frog) . we should have(fox, frog) to prove the transitivity. 2) we need not to check  the missing order to prove transitivity like (bird, bird)] were presented through explanations.

Hope, it clears your doubt.

[Ganesh K Pillai]

Thanks Malabika !!!!

In other words - If I may re-phrase ...

For transitive relation ... out of 3 ordered pairs .. if only one is mentioned ... then we have to assume its Transitive by default ?

If  two out of three is mentioned then ... we need to prove transitivity ... Am I correct with this understanding or is there still some gap ?

I sincerely appreciate your effort in helping me understand . Thanks !!! 

[Ganesh K Pillai]

OR ...

 How about this ?

If only one pair out of three  is mentioned .. we can't prove transitivity so we don't .. we don't have to assume it is transitive ?

If two out of three is mentioned then we have to prove transitivity ?  

Thanks & Regards

K Ganesh

[Malabika Guha Mustafi]

@ Ganesh,

If only one pair out of three  is mentioned .. we can't prove transitivity so we don't .. if only one is mentioned ... then we have to assume its Transitive by default ? Yes.

 In general that's right.

But if any condition is there then we need to check

For example ,

f R= {(x, y}: x is the wife of y},

and we have  elements like ( Hema, Dharmendra) and ( Kareena, Saif ).. The relation is not definitely transitive :) ( Though the missing elements are there).

[Anand Iyer]

If only one out of 3 pairs is available, then it IS transitive.  

If two out of 3 pairs are available, then the condition kicks in.

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Cheers,

Anand

[Ganesh K Pillai]

 Thank you Malabika !!!

 

So it depends on the condition ( if any) , otherwise it could be assumed  to be transitive . Thanks !!!

[Ganesh K Pillai]

My understanding is as follows - 

For injective -for any output  there could be maximum one input  which means there could be a o/p which is not mapped to any pre-image .. condition is maximum one pre-image .

for surjective - for all o/p there has to be minimum one input .

Thanks 

[Ganesh K Pillai]

I have this query from the example of Reflexive relation

mod of a-b <= b .

for reflexive a = b . 

so a-b = a-a =0 .

further it says 0 is not = to b .  

 I am yet to understand ...  this ...

if I put numbers for  a or b ... it satisfies 

for a = 1 

1-1 <= 1 . its less than OR Equal . 0 is not equal but definitely less than 1 .. So thats the confusion ....

Also .. if you can help with some conditions which would make it Reflexive .

Thanks 

K Ganesh

[Maria Joseph]

@Malabika, for the example 2 (for reflexive) I think u have made a mistake or I dont know if I'm.....

Pls check it and if there is a mistake ,post it here .Please

Anyway your work is really appreciating.These days you, @Anand ,@Tejasvi  and more others are helping me a lot.THANKS FOR ALL YOUR WORK....

Regards and God bless u all

Maria

[Malabika Guha Mustafi]

@mjneriam

)A relation R is defined on the set of all real numbers N by ‘a R b’ if and only if |a-b| ≤ b, for a, b ∈ N.

Now when a=b

then |0| not equal to or less than b.

So, the relation is not reflexive. This one?

[Malabika Guha Mustafi]

Thank you  so much for pointing. 

 The explanation should be like that  when a=-2[ here , N is not natural number but defined as real number]

|a-a| =0 

then |0| not equal to or less than b.

Hope, now it clears your doubt.

[Ganesh K Pillai]

Dear Team 
Symmetric relation examples  - a) Line A parallel to Line B , or

 

b) Line A perpendicular to Line B ..

Any  examples to understand Asymmetric and Antisymmetric in similar lines ..

Thanks 

K Ganesh

 

[Malabika Guha Mustafi]

slope, rise/run ..Is n't it antisymmetric relation?

[Anand Iyer]

on the first and third quadrants, it's symmetric.  Thus, when x increases(decreases), y increases(decreases)

But, on the other two quadrants, yes, it's anti-symmetric, unless the slope is -1.  If slope is -1, x = y, and anti-symmetric property doesn't apply.

--

Cheers,

Anand

[Anand Iyer]

https://en.wikipedia.org/wiki/Antisymmetric_relation

look at the example there...good practical one.

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Cheers,

Anand

[Malabika Guha Mustafi]

 If slope is -1, x = y, and anti-symmetric property doesn't apply.

Anti symmetry is not asymmetry. It takes x=y value.

Asymmetry never takes , x=y

[Malabika Guha Mustafi]

on the first and third quadrants, it's symmetric.  Thus, when x increases(decreases), y increases(decreases).

we are talking about antisymmetry not positive or negative relations.

m= -B/A is not symmetry unless A=B .

[Malabika Guha Mustafi]

Sorry m= -A/B

[Anand Iyer]

oh, yes, yes.., my bad.

I'll rather keep shut for some time...making so many mistakes.

--

Cheers,

Anand

[Malabika Guha Mustafi]

No way, it will help us to cope up with exam pressure :). I just love the way you explain things. Very crisp and classy.

[Malabika Guha Mustafi]

@ Anand, Thank you for the example link.

[Anand Iyer]

I love participating in discussions.  I feel so much like I'm in college again...

But, I don't want to mislead anyone.  If it works against my intentions, that's worse than no participating at all...

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Cheers,

Anand

[Malabika Guha Mustafi]

Everyone makes mistakes. We are very clear about your intension. Please participate. :)

[Anand Iyer]

Thanks, Malabika for your nice words...To have people like you who can point out mistakes is great!  Hope our interactions stay the same when the course starts, and through the end of the course too!

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Cheers,

Anand

[Malabika Guha Mustafi]

Hope so. If I can qualify .

[Kaushik Deka]

I think the question is if the (a) relation defined on a set of parallel lines , (b) relation defined on a set of perpendicular lines are symmetric, anti-symmetric and/or asymmetric. The answer is straightforward: both are symmetric, AND NOT (antisymmetric OR asymmetric).   

[Kaushik Deka]

Antisymmetry example for lines: {(a,b) | a is x-intercept, b is y-intercept in the Cartesian Plane}

Asymmetry example for lines: {(a,b) | a is a vertical on negative X-axis ( x < 0 ), b is a vertical on positive X-axis ( x > 0 )}

Hope this helps.

[Kaushik Deka]

The antisymmetry relation above will need another constraint: (x-intercept, y-intercept) pairs for parallel lines !

[Malabika Guha Mustafi]

Very nice explanation.

[Rajashree Govindan]

Sum of all angles of a right angled triangle is 180 deg..however this does not imply that all triangles are right angled triangles..I guess this can be and example of Asymmetric...just guessing..

Regards

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[Malabika Guha Mustafi]

@ Rajeeg, during argument we need to prove for asymmetric that if (a,b) is there then (b,a) should not be a there. for comparison angle should be compared with angles.

See the examples provided by Kaushik(linka.). Anyway, always keep trying. :). Best wishes for exam.

[Rajashree Govindan]

Thanks Malabika for clearing up my concepts😊

[Sanjaykumar]

Most fruitful efforts, @Malabika :)

[Malabika Guha Mustafi]

Who are still little bit confused about power set , cardinality, relation, they may go through this solution. Yesterday , I posted for another thread.

let P be a subset of [(S×S)∖R]. What is the set of possible cardinalities of P? {Activity question from week 1..activity 1.10..question number 7}
It is not power set.

First see what are the elements of |(S × S)\R| 

The elements in SXS:

{(Snail,Snail), (Snail,Fox), (Snail,Bird), (Snail,Frog)

(Fox,Snail),(Fox,Fox), (Fox,Bird),(Fox, Frog ),

(Bird,Snail),(Bird,Fox),(Bird,Bird),(Bird,Frog)

(Frog,Snail), (Frog,Fox),(Frog,Bird),(Frog,Frog)}

Yes, The cardinality is 16 (4X4).

It is not the cardinality of power set means the set of all the subsets of the given set.

 Cardinality of power set  is 2^4=16 but the elements of power set

{{0},{Snail},{Fox}, {Frog},{Bird},{Snail, Fox},{Snail,Frog},{Snail,Bird}, {Fox,Frog},{Fox, Bird} {Bird,Frog},{Snail,Fox, Bird},{Snail,Fox ,Frog},{Fox,Frog ,Bird}, {Snail,Bird,Frog}{ Snail, Fox, Frog, Bird}}.

In this question, we are dealing with cartesian product not power set.

Now 

[(S×S)∖R]     {R={(Snail, Frog), (Bird, Bird), (Fox, Frog), (Snail, Fox)}

 

{(Snail,Snail), (Snail,Fox), (Snail,Bird), (Snail,Frog)

(Fox,Snail),(Fox,Fox), (Fox,Bird),(Fox, Frog),

(Bird,Snail),(Bird,Fox),(Bird,Bird),(Bird,Frog)

(Frog,Snail), (Frog,Fox),(Frog,Bird),(Frog,Frog)}

After removing R part (red one)

The elements in [(S×S)∖R] :

{(Snail,Snail), (Snail,Bird), 

(Fox,Snail),(Fox,Fox), (Fox,Bird),

(Bird,Snail),(Bird,Fox),(Bird,Frog)

(Frog,Snail), (Frog,Fox),(Frog,Bird),(Frog,Frog)}

So, the cardinality=12

Now P is a subset of this set.

so, It may be {} empty set so cardinality 0, 

It may have only one element{(Snail,Snail)} [ it can be any one element] so cardinality 1

It may have any two elements so cardinality will be 2

It may have any three elements so cardinality will be 3

like this way....... cardinality will be 4,5,6,7,8,9,10,11

It may take full set, when P=(S×S)∖R Then The cardinality is 12

Hope, it will help you to clear the concept.

[Isha Roy]

Thank you so much. I'm a 2020 passout with a non maths background and it helped me a lot. @Malabika

[sangi]

Dear All,

Kindly share the solutions for all the problems related to functions with brief explanations. 

Especially  the Activity Question-11  week1 content . 

Thank you.

On Sat, 21 Nov 2020, 06:51 Malabika Guha Mustafi, <malabi...@gmail.com> wrote:

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[Malabika Guha Mustafi]

Sorry, time issue. Please search in forum.You may get all answers.