1. Classification:
- Empty Set (∅): A set with no elements.
- Finite Set: Elements can be counted completely.
- Infinite Set: Elements never end (e.g., Prime numbers).
2. Equal vs Equivalent:
- Equal: Exact same elements.
- Equivalent: Same number of elements (same cardinality).
For a set with n elements:
• Total Subsets = 2n
• Proper Subsets = 2n - 1
1. Is the set {0} an empty set?
Answer: No. It contains one element: 0. An empty set is written as ∅ or { }.
2. If A = {a, b} and B = {1, 2}, are they Equal or Equivalent?
Answer: Equivalent (~). They have the same number of elements (2), but the elements themselves are different.
3. How many subsets does the set {x, y, z} have?
Answer: 8. Using the formula 2n, where n=3, 23 = 2 × 2 × 2 = 8.
4. What is the symbol for a Proper Subset?
Answer: ⊂. This means the subset is smaller than the parent set.
5. Is the set of counting numbers finite or infinite?
Answer: Infinite. Counting numbers (1, 2, 3...) never end.
6. If P = {1, 2, 3} and Q = {3, 1, 2}, are they Equal?
Answer: Yes (=). Order does not matter in sets; they contain the exact same elements.
7. How many proper subsets does a set with 4 elements have?
Answer: 15. Formula: 24 - 1 = 16 - 1 = 15.
8. Define a Universal Set (U).
Answer: A set that contains all the elements under consideration for a specific problem.
9. True or False: ∅ ⊆ A for any set A.
Answer: True. The empty set is a subset of every set.
10. If n(A) = n(B), then A and B must be...
Answer: Equivalent. Same cardinality means they are equivalent.
11. Is {stars in the sky} considered finite or infinite in math?
Answer: Infinite. In school math, it is usually used as an example of an infinite set.
12. True or False: Every set is a subset of itself.
Answer: True. A ⊆ A is always true.
13. List all subsets of {1}.
Answer: ∅ and {1}. Total = 21 = 2.
14. What is the difference between ⊆ and ⊂?
Answer: ⊆ allows the sets to be equal; ⊂ requires the subset to be strictly smaller.
15. Is {x | x is a person over 200 years old} an empty set?
Answer: Yes (∅). Currently, no living human satisfies this.
16. If A ⊆ B and B ⊆ A, then...
Answer: A = B. They must be the exact same set.
17. Identify the set: {2, 4, 6, 8...}.
Answer: Infinite Set. The ellipsis (...) indicates it continues forever.
18. True or False: Equivalent sets are always equal.
Answer: False. {A, B} and {1, 2} are equivalent but NOT equal.
19. How many subsets does ∅ have?
Answer: 1. The only subset of ∅ is ∅ itself (20 = 1).
20. What is a set with a counting process that terminates called?
Answer: Finite Set.