Set of Sets A set can also cont

Question 1

📘 What is a set of sets?

Answer 1

A set that contains other sets as its elements.
✅ Example: A = {{1}, ∅, {1, 2}}

Question 2

📌 What is a power set?

Answer 2

The power set of a set X is the set of all subsets of X.
📘 Notation: P(X)

Question 3

🔢 What is the cardinality of the power set?

Answer 3

A:
If a set has n elements, then:
👉 |P(X)| = 2ⁿ

Question 4

What is |P({2, 3})|?

Answer 4

A:
2 elements → 2² = 4

Question 5

What is |P({1, 2, 3})|?

Answer 5

A:
3 elements → 2³ = 8

Question 6

What is |P({a})|?

Answer 6

A:
1 element → 2¹ = 2

Question 7

List all subsets of A = {2, 3}

Answer 7

A:
P(A) = {∅, {2}, {3}, {2, 3}}

Question 8

List all subsets of C = {a}

Answer 8

A:
P(C) = {∅, {a}}

Question 9

Is {2} the same as 2 in set theory?

Answer 9

A: ❌ No! {2} is a set, but 2 is just a number.

Question 10

Is the empty set ∅ always in a power set?

Answer 10

A: ✅ Yes — the empty set is a subset of every set!

Question 11

Why does 2ⁿ give the number of subsets?

Answer 11

A:
Each element has 2 choices:
✅ In the subset or ❌ Not in the subset
→ So for n elements, 2 × 2 × … n times = 2ⁿ

Question 12

Every element of the power set is also a subset of the original set. T/F

Answer 12

A: ✅ True

Question 13

The power set of {1, 2} includes {1, 2, 3}

Answer 13

A: ❌ False — 3 is not in the original set.

Question 14

If A has 0 elements (A = ∅), what is P(A)?

Answer 14

A:
P(∅) = {∅}
✅ Even the empty set has one subset — itself!