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Belonging and subsets: counting the subsets of a set
Mathematics Prep 1 Term 1, Unit 1, Lesson 3
GoalUse ∈, ∉ and ⊂ correctly, list the subsets of a set and count them.
New words
belongs to ∈ · is an element of a setsubset ⊂ · a set all of whose elements are in another setequal sets · sets with exactly the same elements2ⁿ · the number of subsets of a set with n elements
Explanation
The sign ∈ links an element to a set: 4 ∈ {2, 4, 6}, and 5 ∉ {2, 4, 6}. The sign ⊂ links two sets: A ⊂ B means every element of A is also in B. So {2, 4} ⊂ {2, 4, 6}, but 4 ⊂ {2, 4, 6} is wrong because 4 is an element, not a set. Every set is a subset of itself, and the empty set is a subset of every set. To count the subsets of a set with n elements, think of each element: it is either in or out, two choices each. So the number of subsets is 2 × 2 × … × 2 = 2ⁿ. Two sets are equal when each is a subset of the other.
★Remember
∈ is for an element and a set. ⊂ is for two sets. A set with n elements has 2ⁿ subsets.
Worked examples
A = {2, 4, 6, 8}. Then 4 ∈ A and 5 ∉ A. The set {2, 4} is a subset: {2, 4} ⊂ A. We must not write 4 ⊂ A, because 4 is an element.
| Elements in the subset | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| How many | 1 | 3 | 3 | 1 |
A set with 5 elements. Each element is in or out: 2 × 2 × 2 × 2 × 2 = 2⁵ = 32 subsets. Equal sets: {1, 2, 3} = {3, 2, 1, 1}, because both have exactly the elements 1, 2 and 3.
Class activity
Subset cards. Write 1, 2, 3 on three cards. With a grown-up, make every possible pile (including an empty pile). How many piles are there? Add a fourth card: how many piles now?
Practice
1
Which are true? (a) 3 ∈ {1, 2, 3} (b) 5 ∉ {1, 2, 3} (c) {2} ⊂ {1, 2, 3}
All three are true.
2
Is 2 ⊂ {1, 2, 3} correct? If not, correct it.
No. 2 is an element, so write 2 ∈ {1, 2, 3} or {2} ⊂ {1, 2, 3}.
3
How many subsets does {1, 2, 3, 4} have?
2⁴ = 2 × 2 × 2 × 2 = 16.
4
List all the subsets of {5, 6}.
∅, {5}, {6}, {5, 6}: four subsets (2² = 4).
5
Mina says {1, 2} has 2 subsets, {1} and {2}. What did she forget?
She forgot ∅ and {1, 2} itself. There are 4 subsets.