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Senior Manager  G
Joined: 04 Sep 2017
Posts: 291
The cardinality of a finite set is the number of elements in the set.  [#permalink]

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Difficulty:   95% (hard)

Question Stats: 28% (01:37) correct 72% (01:26) wrong based on 106 sessions

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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.

DS06351.01
VP  D
Joined: 20 Jul 2017
Posts: 1048
Location: India
Concentration: Entrepreneurship, Marketing
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The cardinality of a finite set is the number of elements in the set.  [#permalink]

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gmatt1476 wrote:
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.

DS06351.01

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
##### General Discussion
VP  D
Joined: 19 Oct 2018
Posts: 1071
Location: India
Re: The cardinality of a finite set is the number of elements in the set.  [#permalink]

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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

gmatt1476 wrote:
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.

DS06351.01
Intern  Joined: 20 May 2019
Posts: 3
The cardinality of a finite set is the number of elements in the set.  [#permalink]

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Dillesh4096 wrote:
gmatt1476 wrote:
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.

DS06351.01

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

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!!!
Intern  B
Joined: 07 Jan 2019
Posts: 4
Re: The cardinality of a finite set is the number of elements in the set.  [#permalink]

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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 Re: The cardinality of a finite set is the number of elements in the set.   [#permalink] 08 Nov 2019, 05:25
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