Identifying Subsets & Power Sets (Advanced)

1) If A has n elements, how many elements does P(P(A)) have?

2^(2^n)

2^n

2^(n^2)

N^2

The power set of A has 2^n elements. Taking its power set gives 2^(2^n).

Explanation

The power set of A has 2^n elements. Taking its power set gives 2^(2^n).

2) A has 3 elements. How many elements are in P(P(P(A)))?

2^256

2^128

2^64

2^32

|A|=3 → P(A) has 2^3=8. → P(P(A)) has 2^8=256. → P(P(P(A))) has 2^256.

Explanation

|A|=3 → P(A) has 2^3=8. → P(P(A)) has 2^8=256. → P(P(P(A))) has 2^256.

3) If X ⊂ Y and |Y| = n, how many subsets of Y contain X?

2^(n−|X|)

2^n

N−|X|

2^|X|

Every element not in X can either be included or excluded freely → 2^(n−|X|).

Explanation

Every element not in X can either be included or excluded freely → 2^(n−|X|).

4) How many subsets of a 10-element set have at least one element?

1023

1024

512

1000

Total subsets = 2^10=1024. Excluding ∅ leaves 1023.

Explanation

Total subsets = 2^10=1024. Excluding ∅ leaves 1023.

5) How many proper subsets does a 7-element set have?

127

128

64

126

Total subsets = 2^7=128. Proper subsets exclude the set itself → 127.

Explanation

Total subsets = 2^7=128. Proper subsets exclude the set itself → 127.

6) How many subsets of a 6-element set have exactly 4 elements?

15

10

12

20

Number of 4-element subsets = C(6,4)=15.

Explanation

Number of 4-element subsets = C(6,4)=15.

7) If P(A) has 16 elements, how many elements does A have?

4

8

2

6

|P(A)|=2^|A|=16. So |A|=4.

Explanation

|P(A)|=2^|A|=16. So |A|=4.

8) If |P(A)| = 512, find |A|.

9

8

10

7

2^|A|=512 → |A|=9.

Explanation

2^|A|=512 → |A|=9.

9) For set B={1,2,{3}}, how many subsets does B have?

8

6

9

12

B has 3 elements (1,2,{3}). Subsets=2^3=8.

Explanation

B has 3 elements (1,2,{3}). Subsets=2^3=8.

10) If S has n elements, how many subsets of S have odd cardinality?

2^(n−1)

2^n

N−1

N

Half of all subsets have odd cardinality → 2^(n−1).

Explanation

Half of all subsets have odd cardinality → 2^(n−1).