Subsets: Proper, Improper, and the Formula

Lesson 2 of 4: Sets

In this lesson:

• Decide whether one set is a subset of another
• Distinguish proper from improper subsets
• Derive and apply the formula for the number of subsets

Learning Objectives for This Subsets Lesson

By the end of this lesson, you should:

1. Define a subset and use ⊆ correctly
2. Check whether one set is a subset
3. Distinguish proper from improper subsets
4. Derive the formula for the subset count
5. Apply to find subsets or

Opening Question: How Many Sub-Teams?

A P7 school committee has 3 representatives: {Aisha, Brian, Christine}

How many different sub-teams could be formed — including picking nobody?

• Just Aisha: {Aisha}
• Aisha and Brian: {Aisha, Brian}
• Everyone: {Aisha, Brian, Christine}
• Nobody: { }

How many total? List them all systematically.

Which Candidates Are Valid Sub-Teams?

A = {Aisha, Brian, Christine} is the full committee.

Which candidates are valid sub-teams formed from A only?

T₂ fails: David is not in A.

Defining a Subset of a Set

(read: "A is a subset of B") means:

Every member of A is also a member of B.

• If A has no members (A = ∅), then A ⊆ B automatically
• B is called the superset of A

Subset Notation Versus Membership Notation

Know both symbols:

Membership vs. subset:

• — member
• — subset

Worked Example: Is It a Subset?

P = {1, 2}, Q = {1, 2, 3, 4}. Is P ⊆ Q?

Check each member of P:

• 1 ∈ Q? Yes
• 2 ∈ Q? Yes

Both in Q → P ⊆ Q ✓

Is Q ⊆ P? 3 ∈ P? No → Q ⊄ P

Quick Check: Is It a Subset?

1. Is ?
2. Is ?
3. Is ?

Apply the membership check to each — then advance.

Proper Subsets and the Improper Subset

• Proper subset : A ⊆ B and A ≠ B (A is strictly smaller than B)
• Improper subset when : every set is a subset of itself

Every proper subset is a subset; not every subset is proper.

Worked Example: Proper or Improper?

Classify each relationship:

The Empty Set Is Always a Subset

For any set A,

Why? Is every member of ∅ also in A?

∅ has no members — so no member could fail the test.

The condition is satisfied because nothing can violate it.

, but ∅ still has exactly 1 subset: itself.

Quick Check: List All Subsets of {a, b}

A = {a, b} — list every subset of A:

There are exactly 4. Find them all — then advance.

Build the Subset-Counting Pattern Table

For each set size , list all subsets and count them:

Fill in row 3 yourself before advancing.

The Pattern: Each New Member Doubles the Count

Why does adding one member double the count?

Each existing subset either includes or excludes the new member — so 2 choices per existing subset.

Formula: Number of subsets of a set with members =

Worked Example: Verify with 3-Member Set

A = {p, q, r} — , so expected subsets =

All 8 subsets:

Count: 1 + 3 + 3 + 1 = 8 = ✓

Predict First: Subsets of Ten Members

A set has 10 members.

Is it true that it has 1 024 subsets?

• A. True —
• B. False — the formula doesn't apply here

Commit to your answer before advancing.

Quick Check — Apply the Formula

How many subsets does each set have?

1. A set with 0 members: ?
2. A set with 1 member: ?
3. A set with 4 members: ?
4. A set with 5 members: ?

Finding from the Number of Subsets

Sometimes we work the formula backwards:

If a set has 32 subsets, how many members does it have?

The set has 5 members.

Rule: → ask "2 to which power equals ?"

Worked Example: Reverse the Subsets Formula

Set B has 64 subsets. How many members?

Set C has 16 subsets. How many members? Give an example.

Example: C = {1, 2, 3, 4} has subsets ✓

Your Turn — Full Challenge

Set E = {p, q, r, s} ()

1. How many subsets does E have?
2. List ALL subsets, organised by size (0 members, 1 member, 2 members, etc.)
3. Verify your list matches the formula answer.

Work this completely on your own before advancing.

Real-World Example: Months With 31 Days

Identify the set, then count, then apply the formula.

Set Y = {months with 31 days}:

• Jan, Mar, May, Jul, Aug, Oct, Dec

So , and the number of subsets is:

Total Subsets Versus Proper Subsets

counts ALL subsets — including ∅ and the set itself.

What about only the PROPER subsets?

Exclude the set itself (the improper subset):

• {months with 31 days}: 128 → 127 proper
• {p, q, r}: → 7 proper

Watch Out — Common Errors

✓ Element vs. subset: ≠
✓ ∅ is always a subset:

Error 1: Writing (2 is not a set)
Error 2: Forgetting ∅ → undercount
Error 3: Forgetting the full set → undercount

Key Takeaways From the Subsets Lesson

✓ : every member of A is in B
✓ Proper subset: A ⊂ B, where A ≠ B
✓ ∅ and the set itself are always subsets
✓ members → subsets ( proper)

Next, Lesson 3: Venn diagrams for overlapping sets.