Activity 1.10

[SHIVANI SINGHAL]

In activity q. no is 5 and 7 how to solve ?                                                                                                    Please give me ans.                                                                                                                                                                        Thank You

[Aayushi Dayal]

in question no. 5, there is total 9 elements which are distinct. except for the 2nd option, every other option contains [3,2] or [2,4]. And we need other elements apart from these two.

in question no. 7, S X S has 16 elements. Subtracting R from it, we get 12. possible subsets cannot exceed 12 and it is also possible that there are no elements present in the said subset, so we get the answer,{0,1....12}. 

Hope this helps.

[SHIVANI SINGHAL]

Thanks Aayushi but why option (1,2,....12) is wrong?

[Aayushi Dayal]

In {1,2...12}, the element 0 is missing. in a power set, phi is included i.e, an empty set. the subsets can start from being empty to having 12 elements. i don't know if it'll help you. it' a bit confusing, now that I think of it. all i know is that when we talk about power sets and subsets, empty sets cannot be ignored.

[Malabika Guha Mustafi]

@ Aayushi, you are absolutely right.

Power set means maximum subset can be formed by the elements of a particular set.

So it is starts from empty set.

[subhajit POD]

Hi all,  This is Subhajit from the course instructor team. I am seeing that there is confusion regarding the power set. Power set of a set is the set of all the subsets of the given set. It includes the empty set and the set itself. As the set it self is a subset of that set and the empty set is subset of every set. Hope it clarifies your confusion. 

Best

Subhajit

[ar Memon]

S X S has 16 elements. Subtracting R from it, we get 12. possible subsets cannot exceed 12 and it is also possible that there are no elements present in the said subset,    It includes the empty set and the set itself. As the set it self is a subset of that set and the empty set is subset of every set so now we have total 4 {sutracting  R} + 13 subsets  including 0 total will 17 ........but it should be 16 max

[ar Memon]

please make it clear how total sets 17 possible in above case @ que 7

[ar Memon]

0 should be one of these 16 

[Guganesan S. Ilavarasan]

But in the first place how do we know that S X S is 16? I can't see it being mentioned in the question.

[Malabika Guha Mustafi]

S has 4 elements so SxS  have 16

[Ganesh K Pillai]

Sorry  .. Subhajit ..  I am more confused now ;-) 

Thanks for the effort though . 

On Thursday, November 19, 2020 at 11:48:13 AM UTC+5:30 subh...@onlinedegree.iitm.ac.in wrote:

[ar Memon]

From..... 2^ n so 2^4....malabika is right.  This option contains 0 then 1 to 12 seems wrong because in this way total will be 17 instead of 16,,,,,,,, 13 +4,,,,, malabika please share if you have any logic.....

[Malabika Guha Mustafi]

We have |S| = 4 and |R| = 4.

So, cardinality of the set S × S, |S × S| = |S| × |S| = 4× 4 = 16.

And |(S × S)\R| = 16-4 = 12.

Since P is a subset of (S×S)\R, so cardinality of P can be any natural number less

than or equal to 12. So, possible cardinalities of set P is {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 12}.

Hence, option (d) is correct.

I just copied from team's explanation :). 

[Malabika Guha Mustafi]

let P be a subset of [(S×S)∖R]. What is the set of possible cardinalities of P?
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

In this way ,

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

Hope , it is clear now.

[ar Memon]

Thank you so much.