Subsets and Set Operations

Overview

Recall questions on the difference between a subset and a proper subset, membership tests, and the four set operations.

What you need to know

Answer each one before you reveal it.

Questions from this subtopic

Each one with the model answer underneath.

1. 1

   What does A ⊆ B mean?

   Model answer

   A ⊆ B means A is a subset of B: every element of A is also an element of B. For example {4, 5, 6} ⊆ {4, 5, 6, 7}. The symbol ⊆ includes the case where the two sets are equal, so {4, 5, 6, 7} ⊆ {4, 5, 6, 7} is also true.

2. 2

   What does A ⊂ B mean, and how does it differ from A ⊆ B?

   Model answer

   A ⊂ B means A is a proper subset of B: B contains everything in A, but there is at least one element in B that is not in A, so the two sets cannot be equal. For example {4, 5, 6} ⊂ ℕ, because ℕ contains everything in {4, 5, 6} and much else besides. The difference from ⊆ is exactly that ⊆ includes both ⊂ and equality: {4, 5, 6, 7} ⊆ {4, 5, 6, 7} is true, but {4, 5, 6, 7} ⊂ {4, 5, 6, 7} is false.

3. 3

   Let A = {1, 2, 3, 4} and B = {3, 4, 5}. State whether each of these is true: 2 ∈ A, 5 ∈ A, B ⊆ A, {3, 4} ⊂ B.

   Model answer

   2 ∈ A is true, because 2 is one of the listed elements of A. 5 ∈ A is false; 5 ∉ A. B ⊆ A is false, because 5 is in B but not in A. {3, 4} ⊂ B is true, because B contains both 3 and 4 and also contains 5, which {3, 4} does not, so it is a proper subset.

4. 4

   Name the four set operations and say what each does.

   Model answer

   Membership: tests whether a value is in a set, written x ∈ A, or x ∉ A if it is not. Union: A ∪ B is the set of everything that is in A or in B or in both. Intersection: A ∩ B is the set of everything that is in both A and B. Difference: A \ B, also written A − B, is the set of everything that is in A but not in B.

5. 5

   With A = {1, 2, 3, 4} and B = {3, 4, 5}, work out A ∪ B, A ∩ B, A \ B and B \ A.

   Model answer

   A ∪ B = {1, 2, 3, 4, 5}. A ∩ B = {3, 4}. A \ B = {1, 2}. B \ A = {5}. Note that 3 and 4 appear only once in the union, because each value occurs at most once in a set, and that A \ B and B \ A are different, so difference does not work both ways round.

6. 6

   Give the formal definition of the set difference A \ B.

   Model answer

   A \ B = {x : x ∈ A and x ∉ B} — the set of all x such that x is a member of A and x is not a member of B. It is also written A − B.

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