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Notes for Exercise Set 2.2 - Sets and Notation

Notation and Key Concepts in Sets

  • A set is a collection of distinct elements where order does not matter and duplicates are ignored.
  • Symbols:
    • Membership: "x ∈ A" means x is an element of set A.
    • Real numbers: \mathbb{R} ; Integers: \mathbb{Z}.
  • Set-builder notation: a set is described by a rule, e.g. {x \in \mathbb{R} \mid -1 \le x < 3} means all real numbers x with -1 ≤ x < 3.
  • Interval notation vs set-builder notation:
    • Closed interval: [-1, 3) includes -1 but not 3.
    • Open interval: (-1, 3) excludes both -1 and 3.
  • Cardinality (size) of a set: the number of elements, denoted |A|.
    • Note: duplicates do not count; sets automatically remove duplicates from listings.
  • Nested sets: a set can contain other sets as elements, e.g. {1, {1}, {1, {1}}} has three elements: 1, {1}, and {1, {1}}.
  • Equality of sets:
    • Two sets A and B are equal iff every element of A is in B and every element of B is in A.
    • Order of listing and duplicates do not affect equality.
  • Common notational forms used in this transcript:
    • Real-number set builder: {x \in \mathbb{R} \mid \text{condition}}
    • Integer-set builder: {x \in \mathbb{Z} \mid \text{condition}}
    • Example of an element vs a singleton set: 4 is a number, while {4} is a set with a single element 4.

Exercise Set 2.2 – Key Results and Explanations

  • 1. Which of the following sets are equal?

    • A = {a, b, c, d}

    • B = {d, e, a, c}

    • C = {d, b, a, c}

    • D = {a, a, d, e, c, e}

    • Important observations:

    • Equality ignores order: A and C contain exactly the same elements (a, b, c, d).

    • D has duplicates in its listing, but duplicates do not affect the set; D = {a, d, e, c}.

    • B contains the element e instead of b, so B ≠ A.

    • Conclusions:

    • A = C

    • B = D (since B = {a, c, d, e}, and after removing duplicates, D = {a, c, d, e})

  • 2. How to read each set out loud

    • a. {x \in \mathbb{R} \mid 0 < x < 1}
    • Reading: “the set of real numbers x such that 0 < x < 1.”
    • b. {x \in \mathbb{R} \mid x \le 0 \text{ or } x \ge 1}
    • Reading: “the set of real numbers x such that x is at most 0 or at least 1.”
    • c. {n \in \mathbb{Z} \mid n \text{ is a factor of } 6}
    • Reading: “the set of integers n such that n is a factor (divisor) of 6.”
    • d. {n \in \mathbb{Z} \mid n \text{ is a factor of } 6}
    • Reading: same as (c); note this is a repetition with different typographic cues in the transcript.

  • 3. a–c

    • a. Is 4 = {4}?
    • Answer: No. 4 is a number; {4} is a set containing the number 4. They are not the same object.
    • b. How many elements are in the set {3, 4, 3, 5}?
    • Duplicates are removed in a set, so the elements are {3, 4, 5}; cardinality is |{3,4,3,5}| = 3.
    • c. How many elements are in the set {1, {1}, {1,{1}}}?
    • Elements are: 1, {1}, and {1, {1}}; so cardinality is 3.

  • 4. a–e

    • a. Is 2 \in {2}?
    • Yes. The element 2 is contained in the singleton set {2}.
    • b. How many elements are in the set {2, 2, 2, 2}?
    • Duplicates do not count; the set is {2}; cardinality = 1.
    • c. How many elements are in the set {0, {0}}?
    • Elements are 0 and {0}; cardinality = 2.
    • d. Is {0} = {{0}, {1}}?
    • No. Left side has one element (the set {0}); right side has two elements ({0} and {1}).
    • e. Is 0 = {{0}, {1}}?
    • No. 0 is a number; the right side is a set with two elements; they are not the same object.

  • 5. Which of the following sets are equal?

    • A = {0,1,2}

    • B = {x \in \mathbb{R} \mid -1 \le x < 3}

    • C = {x \in \mathbb{R} \mid -1 < x < 3}

    • D = {x \in \mathbb{Z} \mid -1 < x < 3}

    • E = {x \in \mathbb{Z} \mid -1 < x < 3}

    • Key interpretations:

    • A contains the integers 0, 1, 2.

    • B is the real interval [-1, 3).

    • C is the real interval (-1, 3).

    • D and E are the integers strictly between -1 and 3, i.e. {0, 1, 2}.

    • Equalities:

    • A = D = E, since both sides contain exactly the elements 0, 1, 2.

    • B ≠ C because B includes -1 while C excludes -1; C also excludes -1 and 3 whereas B includes -1 but not 3.

    • B ≠ A and C ≠ A since B and C contain non-integer real numbers not present in A, and A contains only 0,1,2.

  • Summary of key takeaways

    • Order of elements and duplicates do not affect set equality.
    • Set-builder notation and interval notation are two common ways to describe the same collection of numbers, but they may define different collections depending on inclusivity of endpoints.
    • Real vs integer sets can yield different element types; equality depends only on the exact elements, not the description language.
    • Nested sets are legitimate elements, e.g. {1, {1}, {1, {1}}} has three elements but more structure beneath each element.