Mathematical Analysis Course Overview
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Questions and Answers

What does the symbol '∀' represent in logical notation?

  • Belongs to
  • There exists
  • Such that
  • For any (correct)

Which symbol indicates 'there exists' in mathematical logic?

  • ∃ (correct)
  • ∈
  • ∀
  • ⊂

If a statement A implies statement B (𝐴 ⇒ 𝐵), what is condition A called in relation to B?

  • Necessary condition
  • Exclusive condition
  • Equivalent condition
  • Sufficient condition (correct)

What is the correct negation of the statement '∀𝑥 ∈ 𝑋, 𝑥 < 𝑚'?

<p>∃𝑥 ∈ 𝑋 : 𝑥 ≥ 𝑚 (C)</p> Signup and view all the answers

What does the symbol '⇔' signify in logical notation?

<p>Equivalence (C)</p> Signup and view all the answers

What is the correct negation of the statement '∃𝑚 > 0 : ∀𝑥 ∈ 𝑋, |𝑥| ≥ 𝑚'?

<p>∀𝑚 &gt; 0 ∃𝑥 ∈ 𝑋 : |𝑥| &lt; 𝑚 (C)</p> Signup and view all the answers

In the context of the provided information, who is Aleksandr Atvinowski?

<p>Associate Professor of the Department of Mathematical Analysis (D)</p> Signup and view all the answers

Which of the following is the correct way to express 'for any x belonging to the set M' using logical symbols?

<p>∀𝑥 ∈ 𝑀 (C)</p> Signup and view all the answers

Which of the following statements accurately describes the relationship between an element and a set?

<p>An element can either belong to a set or not belong to that set. (C)</p> Signup and view all the answers

How is the empty set typically denoted?

<p>Ø (D)</p> Signup and view all the answers

Given two sets, A and B, what condition must be met for them to be considered equal (A = B)?

<p>Set <em>A</em> and set <em>B</em> must contain the exact same elements. (A)</p> Signup and view all the answers

If set A is a subset of set B (A ⊆ B) , which of the following must be true?

<p>Every element in set <em>A</em> must also be in set <em>B</em>. (D)</p> Signup and view all the answers

What is the crucial difference between a subset and a proper subset?

<p>A proper subset <em>must</em> contain fewer elements; a subset can be equal to the set. (A)</p> Signup and view all the answers

What does the expression ${x \mid x \text{ has the property } P}$ represent?

<p>A set indicated by specifying a characteristic property. (A)</p> Signup and view all the answers

What is the union of sets?

<p>A set containing elements that are in at least one of the sets being considered. (C)</p> Signup and view all the answers

Which of the following demonstrates the transitivity property of set equality?

<p>If <em>A</em> = <em>B</em> and <em>B</em> = <em>C</em>, then <em>A</em> = <em>C</em>. (D)</p> Signup and view all the answers

What does the symbol '↦→' represent in the context of mappings?

<p>A mapping of an element to another element (A)</p> Signup and view all the answers

Given mappings f: X → Y and g: Y → Z, how is the composition of the mappings denoted?

<p>g ∘ f : X → Z (A)</p> Signup and view all the answers

If g is the inverse of mapping f, where is g ∘ f defined and what does it equal?

<p>g ∘ f = IX, the identity map on X (D)</p> Signup and view all the answers

What is a real function of one variable as per the provided definitions?

<p>A mapping f : X → R, where X is a subset of the real numbers (B)</p> Signup and view all the answers

A function where for all $x_1 < x_2$, $f(x_1) < f(x_2)$ or $f(x_1) > f(x_2)$ is called what?

<p>Strictly monotone function (B)</p> Signup and view all the answers

If a function $f : X → R$ is strictly monotonic, what can be inferred about its inverse function $f^{-1}$?

<p>The inverse function exists and is strictly monotonic in the same sense as f (C)</p> Signup and view all the answers

Which of the following is a necessary condition for a function $f : X → R$ to be even?

<p>The domain X must be symmetrical relative to the origin, and f(−x) = f(x) for all x in X (D)</p> Signup and view all the answers

The graph of which type of function is symmetrical with respect to the Y-axis?

<p>Even function (B)</p> Signup and view all the answers

What condition must a function satisfy to be considered odd?

<p>f(−x) = −f(x) and the domain X is symmetric to the origin (B)</p> Signup and view all the answers

Which of the following functions is likely to be an odd function?

<p>$y = x^3$ (A)</p> Signup and view all the answers

Which of the following correctly describes the property of commutativity for the union of two sets?

<p>$A \cup B = B \cup A$ (D)</p> Signup and view all the answers

Given sets A, B, and C, which of the equations below demonstrates the associative property of set union?

<p>$A \cup (B \cup C) = (A \cup B) \cup C$ (A)</p> Signup and view all the answers

What is the result of $A \cap A$?

<p>$A$ (A)</p> Signup and view all the answers

If $A$ is a set and $\emptyset$ is the empty set, what is $A \cap \emptyset$?

<p>$\emptyset$ (D)</p> Signup and view all the answers

What does the distributive property of intersection over union state?

<p>$A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$ (B)</p> Signup and view all the answers

Which of the following is the correct expression for the set difference $A \setminus B$?

<p>{x | x ∈ A and x ∉ B} (A)</p> Signup and view all the answers

Given $A = {1, 3, 5}$ and $B = {4, 5, 6}$, what is $A \cup B$?

<p>$\lbrace 1, 3, 4, 5, 6 \rbrace$ (C)</p> Signup and view all the answers

Using the sets from the previous question, $A = {1, 3, 5}$ and $B = {4, 5, 6}$, what is $A \cap B$?

<p>$\lbrace 5 \rbrace$ (D)</p> Signup and view all the answers

If $A = {p | 0 < p < 30}$ and $B = {p | 10 < p < 40}$, where $p$ are integers, what is $B \setminus A$?

<p>$\lbrace p | 30 \leq p &lt; 40 \rbrace$ (A)</p> Signup and view all the answers

What is the Cartesian product of two sets $A$ and $B$, denoted as $A \times B$?

<p>The set of all ordered pairs $(a, b)$ such that $a \in A$ and $b \in B$. (A)</p> Signup and view all the answers

Flashcards

Set

A collection of distinct objects, often represented by curly braces: {a, b, c}.

Element of a set

An element is a member of a set.

Empty Set (Ø)

The set with NO elements.

Set Equality

Two sets are equal if they contain exactly the same elements.

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Subset (⊆)

All elements of set A are also elements of set B.

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Proper Subset (⊂)

All elements of set A are elements of set B, and A is not equal to B.

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Union (∪)

The set of all elements that are in set A or set B, or both.

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Intersection (∩)

The set of all elements that are in both set A and set B.

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Universal Quantifier (∀)

Used in mathematical statements to indicate that a property holds true for every element in a set.

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Existential Quantifier (∃)

Used in mathematical statements to indicate that at least one element in a set satisfies a specific condition.

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Implication (⇒)

A shorthand way of expressing the logical connection between statements, indicating that the truth of one statement guarantees the truth of another (A implies B).

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Equivalence (⇔)

A shorthand way of expressing that two statements are logically equivalent, meaning they have the same truth value (A is true if and only if B is true).

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Negating Statements with Quantifiers

Finding the negation of a statement involves reversing its truth value, using the appropriate quantifier and changing the property to its opposite.

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Negating Universal Statements

The process of negating a universally quantified statement involves introducing an existential quantifier, changing the statement's property to its opposite, and providing a specific counterexample.

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Negating Existential Statements

The negation of an existentially quantified statement involves introducing a universal quantifier, changing the statement's property to its opposite, and showing that no element satisfies the negated property.

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Negation Symbol (¬)

A negation symbol that changes a statement to its opposite.

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What is the union of sets?

The union of sets A and B is the set containing all elements that are in A, B, or both.

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What is the intersection of sets?

The intersection of sets A and B is the set containing all elements that are in both A and B.

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What is the difference of sets?

The difference of sets A and B is the set containing all elements that are in A but not in B.

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What is a mapping?

A mapping from a set X to Y is a rule that assigns a unique element in Y to each element in X.

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What is the domain of definition?

The domain of definition of a mapping is the set X from which elements are mapped.

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What is the domain of values?

The domain of values of a mapping is the set Y to which elements are mapped.

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What is commutativity?

Commutativity means the order of sets doesn't change the result.

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What is associativity?

Associativity means the grouping of sets doesn't change the result.

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What is the Cartesian product?

The Cartesian product of sets A and B is the set of all possible ordered pairs (a,b) where a is in A and b is in B.

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Explain 𝑓 (𝑥) = 𝑦

The notation 𝑓 (𝑥) = 𝑦 means that the rule 𝑓 maps element 𝑥 from the domain of definition to element 𝑦 in the domain of values.

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Mapping (Function)

A mapping from a set X to a set Y, represented by f: X -> Y, where each element x in X is uniquely mapped to an element y in Y.

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Mapping Graph

The set of all ordered pairs (x, y) where x is an element of X and y is the corresponding element in Y under the mapping f, denoted as Γf = {(x, y) ∈ X x Y | x ∈ X, y = f(x)}.

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Composition of Mappings

A combination of two mappings, where the output of one mapping (f) becomes the input of another mapping (g), written as g∘f: X -> Z, such that for any x in X, (g∘f)(x) = g(f(x)).

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Inverse Mapping

A mapping 𝑔: 𝑌 −→ 𝑋 is the inverse of the mapping 𝑓: 𝑋 −→ 𝑌 if their composition results in the identity mapping: 𝑔∘𝑓 = 𝐼𝑋 and 𝑓∘𝑔 = 𝐼𝑌, where 𝐼𝑋(𝑥) = 𝑥 for all 𝑥 ∈ 𝑋.

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Real Function of One Variable

A real function of one variable, where f maps a subset of real numbers (X ⊂ R) to the set of real numbers.

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Monotonic Function

A function f: X -> R, where X ⊂ R, is called monotone on the set X if the relationship between the input values and the output values follows a specific pattern.

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Even Function

A function f: X -> R, where X ⊂ R, is called even if the set X is symmetric about the origin and ∀𝑥 ∈ 𝑋: 𝑓 (−𝑥) = 𝑓 (𝑥).

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Odd Function

A function f: X -> R, where X ⊂ R, is called odd if the set X is symmetric about the origin and ∀𝑥 ∈ 𝑋: 𝑓 (−𝑥) = −𝑓 (𝑥).

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Strictly Monotone Function

A function f: X -> R, where X ⊂ R, is strictly monotone if it is either strictly increasing or strictly decreasing.

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Inverse of a Strictly Monotone Function

If a function f is strictly monotone, then it has a unique inverse function f⁻¹ that is also strictly monotone.

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Study Notes

Mathematical Analysis Course Information

  • Professor: Aleksandr Atvinowski
  • Department: Mathematical Analysis and Differential Equations
  • Room: 2-7, Building 2
  • Phone: +375 44 702 45 48

Course Literature

  • Konev "Higher Mathematics" textbook and workbook
  • Konev "Limits of Sequences and Functions" textbook and workbook
  • Rudin, Walter, Principles of Mathematical Analysis: International Series in Pure and Applied Mathematics, Bibliography includes index. ISBN: 0-07-054235-X
  • Course in Mathematical Analysis: By Ter-Krikorov A.M., Shabunin M.I.
  • Besov O.V. Lectures on Mathematical Analysis (2 parts)
  • Real and Complex Analysis (6 parts): By Zverovich, Edmund Ivanovitch

Logical Symbolism

  • ∀: Quantifier of generality (any, for any, each)
  • ∃: Quantifier of existence (exists, found)
  • ∈: Belongs to
  • ⊂: Contains
  • ⇒: Implies/following (sufficient)
  • ⇐: Implies/following (necessary)
  • ⇔: Sign of equivalence or equivalence (means A⇒B and B⇒A)

Examples of Logical Symbolism Use

  • ∀x ∈ M: "For any x from the set M"
  • ∃x ∈ M: "There exists x belonging to the set M such that..."

Negation of Statements with Quantifiers

  • Set A: All elements x of the set X satisfy the condition x < m.

    • Negation of A: ∃x ∈ X : x ≥ m

  • Set B: There is a number m > 0 such that all elements x of the set X satisfy the condition |x| ≥ m.

    • Negation of B: ∀m > 0 ∃x ∈ X : |x| < m

Set Theory

  • Elements: x, y, etc.

  • Sets: A, B, etc.

  • Belongs to: ∈ (x ∈ A)

  • Does not belong to: ∉ (x ∉ A)

  • Set: Specifying objects that form the set

  • Empty set: Ø (no elements)

  • Set notation examples:

    • A = {a, b, ..., p}
    • B = {x | x has property P}

  • Equality: Two sets are equal if they contain the same elements (A = B)

  • Subset: Set A is a subset of set B if all elements of A are also elements of B (A ⊆ B)

  • Proper Subset: Set A is a proper subset of set B if A is a subset of B, and A is not equal to B (A ⊂ B)

  • Properties of set equality:

    • Reflexive: A = A
    • Symmetric: A = B, B = A
    • Transitive: A = B and B = C, then A = C

Set Operations

  • Union (∪): The union of sets A and B (A ∪ B) contains all elements in A or B (or both): AUB := {x | or x ∈ A, or x ∈ B, or (x ∈ U and x∈V)}
  • Intersection (∩): The intersection of sets A and B (A ∩ B) contains elements common to both A and B: A ∩ B := {x | x ∈ A and x ∈ B}
  • Difference (): The difference of sets A and B (A \ B) consists of elements in A that are not in B: A\B := {x | x ∈ A and x ∉ B}.

Mappings and Functions

  • Mapping: A rule that maps each element of set X to a single element in set Y. (xy)

  • Domain of definition: The set X

  • Domain of values: The set Y

  • Mapping graph: The set of ordered pairs (x, y) where y= f(x)

  • Composition of mappings: (go f)(x) = g(f(x)).

  • Inverse mapping: g=ƒ−1 such that (go f) = I and (ƒ o g) = I

  • Real function of one variable: a mapping of the form f : X → R, X ⊂ R

Monotonic Functions

  • Increasing: x1<x2 → ƒ(x1) <ƒ(x2)
  • Decreasing: x1<x2 → ƒ(x1) >f(x2)
  • Non-decreasing: x1<x2 → ƒ(x1) ≤ f(x2)
  • Non-increasing: x1<x2 → ƒ(x1) ≥ f(x2)
  • Strict monotonicity: increasing or decreasing

Special Functions

  • Even function: f(-x) = f(x)
    • Example: y = x² , y = cos(x)
  • Odd function: f(-x) = -f(x)
    • Example: y = x,y = sin(x)
  • Periodic function: f(x+T)= f(x) where T is the period.

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