The cardinality of a finite set is the number of elements in the set.

Let 'n' be the cardinality of set A.

(1) 2 is the cardinality of exactly 6 subsets of set A.
Number of 2 sets that can be formed from a set of n elements = nc2 = 6
--> n(n-1)/2 = 6
--> n(n-1) = 12 = 4*3
By comparison, n = 4
So, Cardinality of set A = 4 -- Sufficient


Formula: Number of subsets possible for a set of n elements = \(2^n\) (including the empty set & the set itself)
E.g: If Set A = {1, 2, 3}, All subsets possible = {1}, {2}, {3}, {1,2}, {2,3}, {1,3}, {1,2,3}, {} = 8 subsets i.e, \(2^3\)

(2) Set A has a total of 16 subsets, including the empty set and set A itself.
--> \(2^n\) = 16
--> n = 4
So, Cardinality of set A = 4 -- Sufficient

IMO Option D

germanandres82

When I read "cardinality", I understand basically the number of elements in a set. Following this, my interpretation of the question is if I can know the number of elements in set A. --> Yes, you're understanding of cardinality is fine

There are 6 subsets in set A with 2 elements INSUFFICIENT - THERE COULD BE OTHER SUBSETS IN A OR EVEN SOME OF THE ELEMENTS CAN BELONG TO MULTIPLE SUBSETS --> Not true!!

Can you give any example to prove ??

I will try and explain.

Case 1: Let \(A_1\) has 1 element in it
Assume Set \(A_1\) = {1}
Number of subsets with '0' elements possible = {} --> 1
Number of subsets with '1' element possible = {1} --> 1
--> Total number of subsets possible for \(A_1\) = {}, {1} = 2 or \(2^1\)

Case 2: Let \(A_2\) has 2 elements in it
Assume Set \(A_2\) = {1, 2}
Number of subsets with '0' elements possible = {} --> 1
Number of subsets with '1' element possible = {1}, {2} --> 2
Number of subsets with '2' elements possible = {1, 2} --> 1
--> Total number of subsets possible for \(A_2\) = {}, {1}, {2}, {1, 2} = 4 or \(2^2\)

Case 3: Let \(A_3\) has 3 elements in it
Assume Set \(A_3\) = {1, 2, 3}
Number of subsets with '0' elements possible = {} --> 1
Number of subsets with '1' element possible = {1}, {2}, {3} --> 3
Number of subsets with '2' elements possible = {1, 2}, {2, 3}, {3, 1} --> 3
Number of subsets with '3' elements possible = {1, 2, 3} --> 1
--> Total number of subsets possible for \(A_3\) = {}, {1}, {2}, {3}, {1, 2}, {2, 3}, {3, 1}, {1, 2, 3} = 1 + 3 + 3 + 1 = 8 or \(2^3\)

Case 4: Let \(A_4\) has 4 elements in it
Assume Set \(A_4\) = {1, 2, 3, 4}
Number of subsets with '0' elements possible = {} --> 1
Number of subsets with '1' element possible = {1}, {2}, {3}, {4} --> 4
Number of subsets with '2' elements possible = {1, 2}, {2, 3}, {3, 4}, {4, 1}, {1, 3}, {2, 4} --> 6
Number of subsets with '3' elements possible = {1, 2, 3}, {1, 3, 4}, {2, 3, 4}, {2, 1, 4} --> 4
Number of subsets with '4' elements possible = {1, 2, 3, 4} --> 4
--> Total number of subsets possible for \(A_4\) = {}, {1}, {2}, {3}, {4}, {1, 2}, {2, 3}, {3, 4}, {4, 1}, {1, 3}, {2, 4}, {1, 2, 3}, {1, 3, 4}, {2, 3, 4}, {2, 1, 4}, {1, 2, 3, 4}
= 1 + 4 + 6 + 4 + 1 = 16 or \(2^4\)

NOW THIS IS EXACTLY WHAT WAS GIVEN IN STATEMENT (1), "2 is the cardinality of exactly 6 subsets of set A" = "{1, 2}, {2, 3}, {3, 4}, {4, 1}, {1, 3}, {2, 4} --> 6"
--> It meant exactly 6 subsets of A has cardinality 2. So, We can DEFINITELY say that Cardinality (number of elements) of set A = 4 as per Case 4 --> Sufficient

NOW, Lets consider statement 2
Set A has a total of 16 subsets, including the empty set and set A itself INSUFFICIENT - NO INFORMATION ON NUMBER OF ELEMENTS IN SET A


As per case 4, which has 16 subsets, we can DEFINITELY say that number of elements of set A = 4 (cardinality) --> Sufficient

So, answer is D

Hope it's clear!

Assume the elements in set A= n

Statement 1-
Number of ways to select 2 elements out of n is 6
nC2=6
We can figure out n

Sufficient

Statement 2
nC0+ nC1+nC2+......+nCn=16
\(2^n\)=16

OR

Any element can either present in the subset or not.
Hence total number of subsets- 2*2*2...n times= \(2^n\)

Anyways, we can figure out 'n'

Sufficient

I am honestly completely lost on this one, seems that I have a completely different idea on what the problem is asking.

When I read "cardinality", I understand basically the number of elements in a set. Following this, my interpretation of the question is if I can know the number of elements in set A.

1) 2 is the cardinality of exactly 6 subsets of set A.

There are 6 subsets in set A with 2 elements INSUFFICIENT - THERE COULD BE OTHER SUBSETS IN A OR EVEN SOME OF THE ELEMENTS CAN BELONG TO MULTIPLE SUBSETS

II) Set A has a total of 16 subsets, including the empty set and set A itself INSUFFICIENT - NO INFORMATION ON NUMBER OF ELEMENTS IN SET A

I)-II) INSUFFICIENT - WE KNOW THAT 6 OUT OF THE 16 SUBSETS OF SET A HAVE " ELEMENTS; YET WE DO NOT KNOW HOW MANY ELEMENTS THE OTHER 10 SUBSETS HAVE NOR IF SOME ELEMENTS ARE IN MORE THAN 1 SUBSET

Clearly the question is going in a completely different direction than my line of though, can someone PLEASE help me explain where do I get completely lost?

Thank you!!!

germanandres82

First of all, I believe the first statement could have been structured in a clearer way. First statement actually says: "The number of 2 cardinality subsets in set A is exactly 6", which means there are exactly (and only) 6 two-cardinality subsets in A (no more, no less).

That being said, we can analyze the two statements:

(1) Sufficient. If there are exactly 6 subsets which have cardinality = 2 in set A that means we can make exactly 6 groups of 2 elements out of set A.

Let n be the number of elements in set A. C(n¦2) = n(n-1)(n-2)!/(n-2)!(2!) = 6; from here we have n = 4

(2) Sufficient. Set A has exactly 16 subsets (no more, no less). That means we can make exactly 16 different combinations out of n.

#of combinations we can take out of any set A is: C(n¦0) + C(n¦1) +...+ C(n¦n) = 16. Which n satisfies this condition?

Let's try with n = 1 --> C(1¦0) + C(1¦1) = 2. Nope.

Let's try with n = 2 --> C(2¦0) + C(2¦1) + C(2¦2) = 4. Nope.

Also note that ∑C(n¦k), [with k=0,1,2,...,n] equals 2^n.

Therefore we have: 16 = 2^n from which n=4



Hope that helps

(1) 2 is the cardinality of exactly 6 subsets of set A.
nC2 = 6
n!/(n-2)!*2! = 6
(n)(n-1)/2 = 6
n^2 - n - 12 = 0
n = 4 or -3
n > 0, n = 4
Sufficient

(2) Set A has a total of 16 subsets, including the empty set and set A itself.
2^n = 16
n = 4
Sufficient

Bunuel could you please provide your solution to this question? Thanks!

The cardinality of a finite set is the number of elements in the set. What is the cardinality of set A ?

(1) 2 is the cardinality of exactly 6 subsets of set A.
(2) Set A has a total of 16 subsets, including the empty set and set A itself.

Soln:

Question simply is asking us - How many elements are there in Set A

Lets check each statement one by one.


St(1) 2 is the cardinality of exactly 6 subsets of set A.

Simplifying statement for understanding- st 1 says that - 6 subsets of set A can be formed picking two elements

For example - If there is set A with 3 elements (1,2,3)- i can have subset with no element, second subset with 1 element each , third subset with 2 elements and fourth set of subset with 3 elements in it.

3C0+3C1+3C2+3C3= Total Subset of set A

So for the given set in Question ,lets say it has n elements
St 1 gives us information - that nC2=6

We need to find n here..

To cal n here lets solve
nC2= n!/2! (n-2)!
= n(n-1) (n-2)!/2! (n-2)!

Simplifying can i write n! as n (n-1) (n-2)! [ for understanding- if n=6 ,than 6! can be written as 6*5*4!--same way]

= n(n-1)/2! [(n-2)! cancels out]
=n(n-1)/2 [ we know 2! =2*1=2]

St 1 says nC2= 6
n(n-1)/2= 6

Solving further n(n-1)=12
n^2-n-12=0
n=4 or n=-3

obviously it cannot be -3 , so we know n=4.. Set A has 4 elements. St 1 is sufficient.



(2) Set A has a total of 16 subsets, including the empty set and set A itself.

Total subsets=16 (given)

Remember-Number of subsets for a set of n elements= 2^n

We know 2^n=16
n=4

Sufficient.

Ans is D

Hope this detailed explanation helps!

As written, B should be the answer

(1) 2 is the cardinality of exactly 6 subsets of set A.
All this says is that of 6 subsets of A, the cardinality is 2. This statement doesn't preclude the existence of more subsets though. Clearly this was not the authors intention, however.

(2) Set A has a total of 16 subsets, including the empty set and set A itself.
2^n = 16
n=4

If set A contains duplicates then following example can have also only 6 subsets of 2.
A: 111 22 33
Different possible subsets
11
12
13
22
23
33
But cardinality is 7. The question does not say that elements need to be unique. Can someone please explain.

Posted from my mobile device

are there any similar examples to this type of question. still don't get it after seeing it 3 or 4 times. its just a meh question

here's a simpler way to check cardinality, it's 2^n where n is the number of elements in the set.
We'll check 2nd first..

(2) Set A has a total of 16 subsets, including the empty set and set A itself.

2^n=16 ----- > n=4

(1) 2 is the cardinality of exactly 6 subsets of set A.

I don't usually recommend this, but here, I'm going to do it, I'll use the info from previous statement, so I'll assume there are 4 elements in Set A.

a= {a,b,c,d) "2 is the cardinality of exactly 6 subsets of set A." there are 6 pairs of 2 elements each.

(ab), (ac), (ad), (bc) ,(bd) , (cd) no other pair is possible. (don't use (ab) and (ba) as two pair, they are the same element

Now, is there any other value, apart from n=4 where the # of pairs of 2 is 6. you can try with 6 elements or 8 or 3 but you won't get exactly 6 groups of 2 each. only conceivable way is when n=4.

D is the Answer.

If set A contains duplicates then following example can have also only 6 subsets of 2.
A: 111 22 33
Different possible subsets
11
12
13
22
23
33
But cardinality is 7. The question does not say that elements need to be unique. Can someone please explain.

KarishmaB can you help me out with this question? Tks!

This is a P&C question masquerading as a Sets question. Perhaps that is why it seems confusing.

Cardinality = Number of elements in the set

So say set A has n elements. Then its cardinality is n.
We need to find the number of elements in set A.

(1) 2 is the cardinality of exactly 6 subsets of set A.

When, from set A, we make all subsets, exactly 6 subsets have 2 elements each. Think about it - if you have a set with n elements. How will you make subsets with exactly 2 elements? How many such distinct subsets can you make? In nC2 ways.
So we are given that nC2 = 6
Then n must be 4 because 4C2 = 6. For all greater values of n, nC2 will be greater.
Sufficient alone.

(2) Set A has a total of 16 subsets, including the empty set and set A itself.

From the n elements of set A, total how many subsets can you make? For each element, you can either select it or ignore it. So each element can he handled in 2 ways. IF w have n elements, we get 2*2*2*.. n times
We are given that 2^n = 16
n = 4
Sufficient alone.

Answer (D)

Take an example to help see it clearly. Say set A = {5, 7, 8, 9}
How many subsets can you make with exactly 2 elements? 4C2 = 6. They will include {5, 7}, {5, 8}, {5, 9}, {7, 8}, {7, 9} and {8, 9}
How many total subsets can you make? You can take the 5 or ignore. You can take the 7 or ignore etc. You have subsets ranging from {}, {5}, {7}, ... till {5, 7, 8, 9}
(This is similar to finding total number of factors for a positive integer)
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KarishmaB

The word "exactly" tells us that there are "only" 6 subsets (of A) that have 2 elements, right?

If there is no "exactly" in Statement (1). There could be other subsets of A and Statement (1) will be insufficient.

Not really. "Exactly" only makes the situation crystal clear. It clarifies that when you take all subsets into account, exactly 6 have cardinality 2.
Even if we were given "2 is the cardinality of 6 subsets of set A," we would assume that of the total number of subsets of A, 6 have cardinality 2. This implies that others do not have cardinality 2.

Take another example: 6 of the team members have tested positive.
What does this mean? That others have not. Otherwise, we would have included them too.
If we were awaiting results for others, we would have said "6 of the team members have already tested positive."
When we give an exact number, it pretty much implies it is the EXACT number.
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gmatophobia

Hi,
Can you please help me understand what cardinality is?
Basis what is mentioned in the question, I interpreted it as follows
A = {1,2,3} thus cardinality is 3.
Assuming this as correct interpretation

Statement : 1) 2 is the cardinality of exactly 6 subsets of set A.

There are 6 subsets in set A with 2 elements
Insufficient - There could be other subsets in A or even some of the elements can belong to multiple subsets i.e subsets having cardinality more than or less than 2.

Also, 2 is the cardinality of exactly 6 subsets of set A, I understood it as there will be total of 12 elements with 6 pairs each having 2 elements each

Rickooreo - You have correctly identified the definition of cardinality.

There are 6 subsets in set A with 2 elements

This means if we select any two elements in the set, there are 6 unique subsets.

There may be other subsets with cardinality 3, with cardinality 4 so on. But with cardinality 2, only 6 sub set exists.

In terms of P&C - The statement is similar to saying "the number of ways of selecting two numbers from a set which has n numbers" is 6.

nC2 = 6

n = 4
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