Types of Sets and Subsets

Once you can write a set, the next job is to classify it and compare it with others. This lesson covers the empty set, finite vs infinite, equality, subsets, the power set and the universal set — the language you will use for the rest of the chapter.

Special sets

• Empty (null) set ∅ or { }: contains no element, e.g. {x : x² = −1, x ∈ R}. It is a subset of every set.
• Singleton: exactly one element, e.g. {0}.
• Finite set: countable number of elements; infinite set: unending, e.g. N.

Equal sets

A = B if they have exactly the same elements. {1, 2, 3} = {x : x ∈ N, x ≤ 3}. Repetition and order are ignored.

Subsets

A ⊆ B means every element of A is also in B. If A ⊆ B but A ≠ B, then A is a proper subset, written A ⊂ B. Key facts: ∅ ⊆ A and A ⊆ A always.

Intervals are subsets of R: [a, b] closed, (a, b) open, [a, b) and (a, b] half-open.

Power set and universal set

• Power set P(A): the set of all subsets of A. If A has n elements, P(A) has 2ⁿ elements (and 2ⁿ − 1 proper subsets).
• Universal set U: the fixed set containing all objects under discussion.

Exam Tricks & Tips

• 🎯 A set with n elements has 2ⁿ subsets — the single most-tested fact here.
• 🎯 ∅ is a subset of every set, including itself; but ∅ ≠ {∅} (the latter has one element).
• 🎯 Number of proper subsets = 2ⁿ − 1; non-empty subsets = 2ⁿ − 1 too — read the question carefully.
• 🎯 {a} ⊂ A but a ∈ A — an element and the singleton containing it are different objects.
• 🎯 Equal sets ⇒ same power set; counting elements settles equality fast.
• ❌ Common mistake: treating ∅ and {∅} as identical. ∅ has 0 elements; {∅} has 1 element (namely ∅).

Expected exam pattern

Counting subsets/power-set size (2ⁿ), identifying proper subsets, and matching interval notation to set-builder form are the standard 1-mark items.

Cover the answer, recall, then check. 12 cards on set classification and subsets.

Q1. What is the empty set and its symbol?
A1. A set with no elements, written ∅ or { }.

Q2. Is ∅ a subset of every set?
A2. Yes — the empty set is a subset of every set (including itself).

Q3. How many elements are in {∅}?
A3. One — its single element is ∅. So ∅ ≠ {∅}.

Q4. When are two sets equal?
A4. When they contain exactly the same elements (order and repetition ignored).

Q5. Define subset A ⊆ B.
A5. Every element of A is also an element of B.

Q6. Difference between ⊆ and ⊂ (proper subset)?
A6. A ⊂ B means A ⊆ B and A ≠ B.

Q7. If A has n elements, how many subsets does it have?
A7. 2ⁿ subsets.

Q8. How many proper subsets does an n-element set have?
A8. 2ⁿ − 1.

Q9. What is a power set P(A)?
A9. The set of all subsets of A; it has 2ⁿ elements.

Q10. How many subsets does {a, b, c} have? List the count.
A10. 2³ = 8 subsets.

Q11. What does the closed interval [a, b] mean as a set?
A11. {x : a ≤ x ≤ b, x ∈ R}.

Q12. What is the universal set?
A12. The fixed set U containing all objects relevant to the discussion.

Not all sets are created equal

Once you can write a set, the next question is how sets relate to one another and to their own size. This topic hands you the vocabulary — subset, power set, universal set — used in every proof for the rest of the year.

Core idea: Sets are classified by (a) how many elements they contain and (b) whether one sits inside another.

Classifying by size

Beginner

• Empty (null) set ∅ or { }: has no elements, e.g. {x : x² = −1, x ∈ ℝ}. There is exactly one empty set.
• Singleton set: exactly one element, e.g. {0}.
• Finite set: counting stops, e.g. {vowels}. Its cardinality n(A) is the number of elements.
• Infinite set: counting never stops, e.g. ℕ.

⚠ ∅ and {∅} are different: ∅ is empty; {∅} is a singleton whose sole element is the empty set, so n({∅}) = 1.

Relationships between sets

Intermediate: subsets

A ⊆ B means every element of A is also in B. If additionally A ≠ B, then A is a proper subset, written A ⊂ B.

• ∅ is a subset of every set.
• Every set is a subset of itself.
• Equal sets: A = B ⟺ A ⊆ B and B ⊆ A (this double-inclusion is the standard way to prove two sets are equal).

The universal set U is the fixed "background" set containing all objects under discussion in a problem.

Advanced: the power set

The power set P(A) is the set of all subsets of A. If n(A) = n, then:

n(P(A)) = 2ⁿ (number of subsets)
Number of proper subsets = 2ⁿ − 1

Why 2ⁿ? Building a subset means making an independent yes/no choice ("include or not") for each of the n elements → 2 × 2 × … × 2 = 2ⁿ.

Worked example

Q. A = {1, 2, 3}. List P(A) and state n(P(A)).
Solution. Subsets by size:

• size 0: ∅
• size 1: {1}, {2}, {3}
• size 2: {1,2}, {1,3}, {2,3}
• size 3: {1,2,3}
  So P(A) = {∅, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}}, and n(P(A)) = 2³ = 8. ✓

Where it shows up

Every possible "menu combination", every subset of features in a product tier, every switch-on/switch-off configuration of n devices is counted by 2ⁿ.

Exam tricks & shortcuts

• Number of subsets containing exactly r elements = nCr. Summing gives 2ⁿ.
• Mnemonic: "Power of a set is 2-to-the-power of its size."
• For "proper subsets" subtract 1; for "proper subsets excluding ∅" subtract 2.

Key formulas

• Subset: A ⊆ B if every element of A is in B; proper subset A ⊂ B.
• Number of subsets of a set with n elements: 2ⁿ.
• Number of proper subsets: 2ⁿ − 1; non-empty subsets: 2ⁿ − 1.
• Power set P(A): the set of all subsets; n(P(A)) = 2ⁿ.
• Equal sets: A = B if A ⊆ B and B ⊆ A.
• Intervals: [a,b], (a,b), etc., are subsets of ℝ.

The empty set is a subset of every set; a set with n elements has 2ⁿ subsets.

Worked Example

Problem: A set A has 4 elements. Find (a) the total number of subsets and (b) the number of proper subsets of A.

Solution:
Step 1 — Recall the rule: a set with n elements has exactly 2ⁿ subsets (this counts every possible selection, including the empty set and the set itself).

Step 2 — Here n = 4, so the total number of subsets is:
2ⁿ = 2⁴ = 16.

Step 3 — Recall that a proper subset is any subset except the set itself. So we exclude the one subset that equals A:
Number of proper subsets = 2ⁿ − 1 = 16 − 1 = 15.

Step 4 — Interpretation: the 16 subsets include the empty set ∅ (which is a subset of every set) and A itself; removing only A gives the 15 proper subsets.

Answer: (a) 16 subsets; (b) 15 proper subsets.