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Worksheet

A6.3 Sets

Theory of computation · A level · AQA 7517 4.4.2.2 · about 20 min

BugBotLab
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What this lesson is about

Set notation and comprehension, finite and countably infinite sets, cardinality, Cartesian product, subsets, union, intersection and difference.

Questions 6 marks in all

  1. [1 mark]A = {3, 5, 7} and B = {a, b, c, d}. What is |A × B|?

  2. [1 mark]A = {1, 2, 3, 4} and B = {3, 4, 5}. List A \ B (the difference), as numbers separated by commas in ascending order.

  3. [1 mark]Which set is {2x | x ∈ ℕ ∧ x < 4}?

    1. A{0, 2, 4, 6}
    2. B{2, 4, 6, 8}
    3. C{0, 1, 2, 3}
    4. D{2, 4, 6}

  4. [1 mark]Which of these sets are countably infinite?

    Tick every answer that is true.

    1. AThe natural numbers ℕ
    2. BThe integers ℤ
    3. C{x | x ∈ ℕ ∧ x < 1000}
    4. DThe set of all binary strings

  5. [1 mark]A = {1, 3, 5}. Which statement is false?

    1. A{1, 5} ⊂ A
    2. BA ⊆ A
    3. CA ⊂ A
    4. D{} ⊆ A

  6. [1 mark]What does this program print?

    A = {1, 3, 7, 9}
B = {2, 3, 9}
print(sorted(A | B))
print(sorted(A & B))
print(len(A - B), len(B - A))

The task: two robots' sightings

Two robots patrolled the same room and logged the marker ids they saw, with repeats. Let A be the set of ids robot A saw and B the set robot B saw. - Start from the two lists in the starter. Do not type any of the answers. - Print exactly these eight lines, in this order. A list of ids is written in ascending order separated by single spaces: 1. A: then the members of A 2. B: then the members of B 3. union: then the members of A ∪ B 4. intersection: then the members of A ∩ B 5. A minus B: then the members of A \ B 6. pairs: then the cardinality of A × B, as a whole number 7. subset: then True or False: whether {7, 9} is a proper subset of A 8. even: then the members of {x | x ∈ A ∪ B ∧ x is even}, built with a set comprehension For example, if A were {2, 5} the first line would be A: 2 5.

# the two lines every program starts with: the commands, then the robot
from bugbot import *
connect()

seen_by_a = [3, 7, 1, 7, 9, 3]
seen_by_b = [7, 2, 9, 9, 4]

Plan your program here, then type it in and press Run.

QR code
Do it on the robot
www.bugbotlab.com/learn/a6-3-sets/
The simulator checks it and tells you when it passes. Nothing to install, no account.

Challenges

  1. Write {x² | x ∈ ℕ ∧ x < 6} by listing its members, then check with a Python set comprehension.
  2. What is |A × B × C| if |A| = 3, |B| = 2 and |C| = 4? Why?
  3. Is {x | x ∈ ℤ ∧ x < 0} finite, countably infinite, or neither? Show how to count it off.