Podcast
Questions and Answers
Which of the following functions could qualify as a metric on a non-empty set $X$?
Which of the following functions could qualify as a metric on a non-empty set $X$?
- A function $d: X \times X \to \mathbb{R}$ that satisfies $d(x, y) = d(y, x)$ and $d(x, y) \leq d(x, z) + d(z, y)$ for all $x, y, z \in X$, but allows negative values.
- A function $d: X \times X \to \mathbb{R}$ such that $d(x, y) > 0$ if $x \ne y$, and $d(x, x) = 0$ for all $x, y \in X$.
- A function $d: X \times X \to \mathbb{R}$ that satisfies $d(x, y) \geq 0$, $d(x, y) = 0 \iff x = y$, $d(x, y) = d(y, x)$, and $d(x, y) \leq d(x, z) + d(z, y)$ for all $x, y, z \in X$. (correct)
- A function $d: X \times X \to \mathbb{R}$ such that $d(x, y) = -d(y, x)$ for all $x, y \in X$.
Suppose $(X, d)$ is a metric space. If $d(a, b) = 0$, what can be definitively concluded about $a$ and $b$?
Suppose $(X, d)$ is a metric space. If $d(a, b) = 0$, what can be definitively concluded about $a$ and $b$?
- $a$ and $b$ are distinct elements in $X$.
- $a$ and $b$ are the same element in $X$. (correct)
- $a$ and $b$ are related, but not necessarily equal.
- No conclusion can be made about the relationship between $a$ and $b$ without additional information.
Consider a set $S = {x, y, z}$ and a function $d: S \times S \to \mathbb{R}$ defined as follows: $d(x, y) = 1$, $d(y, z) = 2$, $d(x, z) = 3$, and $d(a, a) = 0$ for all $a \in S$. Also, $d(a,b) = d(b,a)$. Which metric space property does this function violate, if any?
Consider a set $S = {x, y, z}$ and a function $d: S \times S \to \mathbb{R}$ defined as follows: $d(x, y) = 1$, $d(y, z) = 2$, $d(x, z) = 3$, and $d(a, a) = 0$ for all $a \in S$. Also, $d(a,b) = d(b,a)$. Which metric space property does this function violate, if any?
- Triangle inequality (correct)
- Symmetry
- Non-negativity
- Identity of indiscernibles
Let $X$ be a set and $d$ be a metric on $X$. If for some distinct points $x, y, z \in X$, $d(x, y) + d(y, z) = d(x, z)$, what does this imply geometrically?
Let $X$ be a set and $d$ be a metric on $X$. If for some distinct points $x, y, z \in X$, $d(x, y) + d(y, z) = d(x, z)$, what does this imply geometrically?
Which of the following best describes a 'well-defined' set?
Which of the following best describes a 'well-defined' set?
Given sets A, B, and C, which of the following expressions correctly applies the distributive law?
Given sets A, B, and C, which of the following expressions correctly applies the distributive law?
If set A is a proper subset of set B, which of the following statements must be true?
If set A is a proper subset of set B, which of the following statements must be true?
Given a universal set U and a set A, what is the result of $A \cup A^c$?
Given a universal set U and a set A, what is the result of $A \cup A^c$?
Let $A = {1, 2, 3}$ and $B = {3, 4, 5}$. What is $A - B$?
Let $A = {1, 2, 3}$ and $B = {3, 4, 5}$. What is $A - B$?
According to De Morgan's Laws, what is the complement of the union of two sets A and B, i.e., $(A \cup B)^c$?
According to De Morgan's Laws, what is the complement of the union of two sets A and B, i.e., $(A \cup B)^c$?
If $A = {x | x \in \mathbb{R}, 0 < x < 2}$ and $B = {x | x \in \mathbb{R}, 1 < x < 3}$, what is $A \cap B$?
If $A = {x | x \in \mathbb{R}, 0 < x < 2}$ and $B = {x | x \in \mathbb{R}, 1 < x < 3}$, what is $A \cap B$?
Consider an indexed family of sets ${A_\alpha}{\alpha \in \Lambda}$. Which expression correctly represents $(\bigcap{\alpha \in \Lambda} A_\alpha)^c$ according to De Morgan's Laws?
Consider an indexed family of sets ${A_\alpha}{\alpha \in \Lambda}$. Which expression correctly represents $(\bigcap{\alpha \in \Lambda} A_\alpha)^c$ according to De Morgan's Laws?
Let $U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}$ and $A = {2, 4, 6, 8, 10}$. What is $A^c$?
Let $U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}$ and $A = {2, 4, 6, 8, 10}$. What is $A^c$?
Flashcards
Metric
Metric
A function that defines the distance between elements in a set.
Metric Space (X, d)
Metric Space (X, d)
A set X with a distance function d(x, y) that satisfies the metric properties.
d(x, y) ≥ 0
d(x, y) ≥ 0
Distance between any two points is always non-negative.
d(x, y) = 0 iff x = y
d(x, y) = 0 iff x = y
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Set
Set
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Proper Subset
Proper Subset
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Union of Sets (A ∪ B)
Union of Sets (A ∪ B)
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Intersection of Sets (A ∩ B)
Intersection of Sets (A ∩ B)
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Set Difference (A - B)
Set Difference (A - B)
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Complement of A (Aᶜ)
Complement of A (Aᶜ)
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Commutative Property (Sets)
Commutative Property (Sets)
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Associative Property (Sets)
Associative Property (Sets)
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De Morgan's Laws (Sets)
De Morgan's Laws (Sets)
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Study Notes
- Metric Spaces studied by P.K. Jain & Khalil Ahmed
Metric and Metric Spaces
- Metric is a distance function.
- (X,d) is a metric space if it satisfies the following conditions for all x, y in X:
- d(x,y) ≥ 0
- d(x,y) = 0 if and only if x = y
- d(x,y) = d(y,x)
- d(x,z) ≤ d(x,y) + d(y,z)
- d maps X × X to ℝ.
Set Theory Preliminaries
- A set is a well-defined, unordered collection of distinct objects, also referred to as an aggregate, class, or family.
- Tabulated form example: A = {a, e, i, o, u}
- Defining property method example: A = {x | x is a vowel in English}
- a ∈ A means "a" is an element of "A"
- a ∉ A means "a" is not an element of "A"
- ∅ represents the empty set
- ∅ ⊆ A: The empty set is a subset of every set A
- A ⊆ A: Every set A is a subset of itself
Subsets and Proper Subsets
- A ⊆ B: A is a subset of B if x ∈ A implies x ∈ B.
- Proper Subset: A is a proper subset of B if A ≠ ∅, A ≠ B, and A ⊆ B, denoted as A ⊊ B.
- B = {1, 2, 3, 4} and A = {1, 2}, A is a proper subset of B.
Set Operations
Union
- A ∪ B = {x | x ∈ A or x ∈ B}
- Represents all elements in A or B
Intersection
- A ∩ B = {x | x ∈ A and x ∈ B}
- Represents all elements in both A and B
Set Difference
- A - B = {x | x ∈ A but x ∉ B}
- Elements that are in A but not in B
- B - A = {x | x ∈ B but x ∉ A}
- A - B ≠ B - A
Complement
- Ac = U - A, where U is the universal set.
- Ac represents all elements in U that are not in A.
- A - B = {x ∈ U | x ∈ A but x ∉ B} = A ∩ Bc
Laws Governing Sets
Commutative Property
- A ∪ B = B ∪ A and A ∩ B = B ∩ A
Associative Property
- A ∪ (B ∪ C) = (A ∪ B) ∪ C
- A ∩ (B ∩ C) = (A ∩ B) ∩ C
Identity Laws
- A ∪ U = U
- A ∩ U = A, where U is the universal set.
Complement Laws
- A ∪ Ac = U
- A ∩ Ac = ∅
Distributive Laws
- A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
- A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
De Morgan's Laws
- (A ∪ B)c = Ac ∩ Bc
- (A ∩ B)c = Ac ∪ Bc
Operations on Finitely Many Sets
- A₁, A₂, ..., Aₙ are finitely many sets.
- ⋃ᵢ<binary data, 1 bytes><binary data, 1 bytes><binary data, 1 bytes>₁ⁿ Aᵢ = {x | x ∈ Aᵢ for at least one i}
- ⋂ᵢ<binary data, 1 bytes><binary data, 1 bytes><binary data, 1 bytes>₁ⁿ Aᵢ = {x | x ∈ Aᵢ for each i}
- {1, 2, 3, ..., n} is the indexing set.
Operations on an Indexing Set
- Indexed Set: N = {1, 2, 3, 4, ...}
- ⋃ᵢ<binary data, 1 bytes><binary data, 1 bytes><binary data, 1 bytes>₁^∞ Aᵢ = {x | x ∈ Aᵢ for some i ∈ N}
- ⋂ᵢ<binary data, 1 bytes><binary data, 1 bytes><binary data, 1 bytes>₁^∞ Aᵢ = {x | x ∈ Aᵢ for each i ∈ N}
- Let Λ be an indexing set, {Aα}α∈Λ be an indexed family of sets.
- ⋃α∈Λ Aα = {x | x ∈ Aα for some α ∈ Λ}
- ⋂α∈Λ Aα = {x | x ∈ Aα for each α ∈ Λ}
DeMorgan's laws
- (⋃α∈Λ Aα)ᶜ = ⋂α∈Λ Aαᶜ
- (⋂α∈Λ Aα)ᶜ = ⋃α∈Λ Aαᶜ
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Description
Brief notes on metric spaces including the definition of a metric and conditions for a metric space. Also covers set theory basics: set notation, elements, the empty set, subsets, and proper subsets.
