Set Theory Quiz

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Questions and Answers

The intersection of a family of sets (Ai)i∈I is denoted by "______"

∩i∈I Ai

The union of a family of sets (Ai)i∈I includes all elements that belong to at least one set in the family.

True (A)

What is the difference between the intersection and the union of a family of sets?

The intersection of a family of sets includes elements that are common to all sets in the family. The union includes all elements that are in at least one set in the family.

Which of the following statements is TRUE about the intersection of a family of sets (Ai)i∈I?

An element belongs to the intersection if it belongs to all Ai. (C) Signup and view all the answers

Match the following symbols with their corresponding operations on families of sets:

∩i∈I Ai = Intersection ∪i∈I Ai = Union ∃ i ∈ I, x ∈ Ai = Belongs to at least one set ∀ i ∈ I, x ∈ Ai = Belongs to all sets Signup and view all the answers

In the example given, A1 = {1, 2, 3, 4,...}, A2 = {2, 4, 6, 8,...}, ... . The intersection of all these sets, denoted by ∩i∈I Ai, is the set {12, 24, 36, 48,...}. This means that the intersection contains all ______ of all sets Ai.

common multiples Signup and view all the answers

The union of sets includes elements that are in exactly one set.

False (B) Signup and view all the answers

Explain the meaning of the symbol '∃' in the context of sets?

The symbol '∃' represents the existential quantifier and means 'there exists'. It indicates that there is at least one element that satisfies a given condition. Signup and view all the answers

What is the term used to describe the set of all mappings from A to B?

B^A (B) Signup and view all the answers

A correspondence is a mapping if and only if for every element of A, there is only one corresponding element in B.

True (A) Signup and view all the answers

What is the notation used to represent a mapping 'f' from A to B?

f: A -> B Signup and view all the answers

The element in B that corresponds to an element x in A, by the mapping f, is called the ______ of x.

image Signup and view all the answers

Match the following terms with their corresponding definitions:

Correspondence = An ordered triple (A, B, G) where G is a subset of A × B. Codomain = The set B in a correspondence (A, B, G). Graph = The set G in a correspondence (A, B, G). Domain = The set A in a correspondence (A, B, G). Signup and view all the answers

If (x, y) belongs to the graph G of a correspondence C from A to B, what does it signify?

x is an element of A and y is an element of B. (A) Signup and view all the answers

In a correspondence, an element from the domain can have multiple corresponding elements from the codomain.

True (A) Signup and view all the answers

What term is used to describe the element of A in a correspondence (A, B, G)?

Domain Signup and view all the answers

What is the notation used to represent the direct image of a subset X under a mapping f?

f(X) (C) Signup and view all the answers

If a ∈ A and f(a) ∈ f(X), then a must be an element of X.

False (B) Signup and view all the answers

What is the direct image of the set X = {-2, -1, 1, 2} under the mapping f(x) = x²?

{1, 4} Signup and view all the answers

If X is an empty set, then the direct image of X under any mapping f is also an ______ set.

empty Signup and view all the answers

Which of the following statements is TRUE regarding the direct images of subsets under a mapping?

The direct image of the union of two sets is equal to the union of their direct images. (C) Signup and view all the answers

Match the following properties of mappings with their corresponding definitions:

f(X) = ∅ ⇐⇒ X = ∅ = The direct image of an empty set is empty, and vice versa X ⊆ X' ⇒ f(X) ⊆ f(X') = If a subset is contained within another, its direct image is contained within the direct image of the larger set f(X ∪ X') = f(X) ∪ f(X') = The direct image of the union of two sets is the union of their direct images f(X ∩ X') ⊆ f(X) ∩ f(X') = The direct image of the intersection of two sets is included in the intersection of their direct images Signup and view all the answers

In the context of a magma (E, >), an element e of E is an identity for > if and only if γe = δe = ______

IdE Signup and view all the answers

According to the provided content, a magma can have multiple identity elements.

False (B) Signup and view all the answers

What is the condition for an element a in a magma (E, >) to be considered idempotent for >?

a>a = a Signup and view all the answers

Which of the following statements accurately describes the concept of a left invertible element 'a' in a unitary magma (E, >) with identity element 'e'?

There exists an element 'a0' in 'E' such that 'a0 > a = e'. (A) Signup and view all the answers

In the monoid (N, +), every element is invertible.

False (B) Signup and view all the answers

Match the following terms with their appropriate definitions:

Magma = A set with a binary operation Monoid = An associative and unitary magma Identity element = An element e in a magma such that e>x = x>e = x for all x in the magma Idempotent element = An element a in a magma such that a>a = a Invertible element = An element a in a magma that has an inverse Signup and view all the answers

Which of the following statements is true regarding the mapping f: A -> B and a family of subsets (Yi)i∈I of B?

f^(-1)(∩i∈I Yi) = ∩i∈I f^(-1)(Yi) (A), f^(-1)(∪i∈I Yi) = ∪i∈I f^(-1)(Yi) (C) Signup and view all the answers

For any subset X of A, f(X) is always a subset of f^(-1)(f(X))?

False (B) Signup and view all the answers

In Proposition 5.5, what is the implication of the statement '∀ X ∈ P(A), we have X ⊆ f^(-1)(f(X))'?

It implies that the pre-image of the image of a set X under f is always a superset of X. Signup and view all the answers

A mapping f: A -> B is ______ if and only if for any subset X of A, f^(-1)(f(X)) = X.

injective Signup and view all the answers

Match the following conditions with their corresponding property of a mapping:

f is injective = ∀ X ∈ P(A), f^(-1)(f(X)) = X f is surjective = ∀ Y ∈ P(B), f(f^(-1)(Y)) = Y f is bijective = f is both injective and surjective Signup and view all the answers

Which of the following is NOT a condition equivalent to f being injective?

∀ Y ∈ P(B), f(f^(-1)(Y)) = Y (D) Signup and view all the answers

If f(X ∩ X') = f(X) ∩ f(X') for all X, X' ∈ P(A), then f is necessarily injective.

True (A) Signup and view all the answers

Describe the relationship between the surjectivity of a mapping f: A -> B and the image of the preimage of a subset Y of B.

If f is surjective, then the image of the preimage of Y under f is equal to Y itself. In other words, every element in Y has at least one preimage in A. Signup and view all the answers

Which of the following is NOT a characteristic of a mapping?

An element in the domain can have multiple images in the codomain. (D) Signup and view all the answers

The correspondence defined in Example 1.1 is a mapping.

False (B) Signup and view all the answers

What is the name given to the mapping that maps each element of a set to itself?

Identity mapping Signup and view all the answers

The image or range of a mapping f is denoted by ______.

Imf Signup and view all the answers

Match the following mappings with their descriptions:

IdA = Identity mapping of A pr1 = First projection of A × B pr2 = Second projection of A × B f = Restriction of f to X Signup and view all the answers

Two mappings f and g are equal if they have the same domain and codomain.

False (B) Signup and view all the answers

What is the restriction of a mapping f to a subset X of its domain?

A mapping that maps elements of X to their corresponding images under f. Signup and view all the answers

The composition of two mappings f: A → B and g: B → C is possible if:

The codomain of f is equal to the domain of g. (A) Signup and view all the answers

Study Notes

Logic and Theory of Sets

Logic: A tool for reasoning about the truth and falsity of mathematical statements.
Statement/Proposition: An assertion that is either true or false, but never both simultaneously.
- Example: "1 + 0 = 1" is a true statement.
- Example: "x≤ 9" is not a statement as x is not specified.
Truth Value: Assigned to a statement, denoted by T (true) or F (false).
Compound Statements: Formed by joining statements using logical connectors.
Logical Connectors:
- Negation (˥P):
  - True when P is false.
  - False when P is true.
- Conjunction (P∧Q):
  - True when both P and Q are true.
  - False otherwise.
- Disjunction (PVQ):
  - False when both P and Q are false.
  - True otherwise.
- Implication (P⇒Q):
  - False when P is true and Q is false.
  - True otherwise.
- Equivalence (P⇔Q):
  - True when P and Q have the same truth value.
  - False otherwise.
Tautology: A compound statement that is always true, regardless of the truth values of its component statements.
Contradiction: A compound statement that is always false, regardless of the truth values of its component statements.
Principle of Non-Contradiction: The statement P ∧ ˥P is a contradiction.
Law of the Excluded Middle: The statement P ∨ ˥P is a tautology.

Sets

Set: An unordered collection of distinct objects (elements/members).
Set Membership: x ∈ A, means x is an element of A. x ∉ A means x is not an element of A.
Extensional Definition of a Set: Listing all elements within braces.
Intentional Definition of a Set: Defining the elements via a property.
- Example: {x ∈ Ν / 1 ≤ x ≤ 7}
Singleton: A set with exactly one element.
Doubleton: A set with exactly two elements.
Tripleton: A set with exactly three elements.
Empty Set: A set with no elements (denoted by Ø).
Subset (A ⊆ B): Every element in A is also in B.
Proper Subset (A ⊂ B): A ⊆ B and A ≠ B (A is a subset of B but not equal to B).
Power Set (P(A)): The set of all subsets of A.

Operations on Sets

Intersection (A ∩ B): Elements belonging to both A and B.
Union (A ∪ B): Elements belonging to A or B (or both).
Disjoint Sets (A ∩ B = Ø): Sets with no common elements.
Relative Complement (CSA or A – B): Elements in S but not in A.
Equality of Sets (A = B): If two sets A and B have the same elements.

Mappings/Functions

Correspondence: A relationship from a set A to a set B.
Mapping/Function: A correspondence from A to B where, for every element of A, there is a unique corresponding element in B.
Image (f(x)): The element in B associated with x in A.
Preimage: elements in A which correspond to a given element in B
Restriction of a function: A function defined only on a subset of the domain
Identity Mapping (IdA): Each element maps to itself.
Image or range of f (Imf): The set of all images of elements in A.
Injective (Injection): Each element in the codomain has at most one preimage in the domain
Surjective (Surjection): Each element in the codomain has at least one preimage in the domain
Bijective: Both injective and surjective
Composite Mapping (go f): The output of g applied to the output of f.

Subsets and Power Set

Subset: A set A is a subset of a set B if every element of A is also an element of B.
Proper subset: A is a proper subset of B if A is a subset of B and A is not equal to B.
Power set: the power set of a set is the set of all subsets of that set.

Methods of Proof

Direct Proof: Assume the premise is true, and show that the conclusion is also true.
Proof by Contraposition: Prove the contrapositive of the implication.
Proof by Contradiction: Assume the negation of the statement is true, and derive a contradiction.

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Test your knowledge on the concepts of intersection and union of sets in Set Theory. This quiz covers fundamental operations on families of sets, their symbols, and definitions. Perfect for students looking to deepen their understanding of this important mathematical concept.

Set Theory Quiz

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Questions and Answers

The intersection of a family of sets (Ai)i∈I is denoted by "______"

The union of a family of sets (Ai)i∈I includes all elements that belong to at least one set in the family.

What is the difference between the intersection and the union of a family of sets?

Which of the following statements is TRUE about the intersection of a family of sets (Ai)i∈I?

Match the following symbols with their corresponding operations on families of sets:

In the example given, A1 = {1, 2, 3, 4,...}, A2 = {2, 4, 6, 8,...}, ... . The intersection of all these sets, denoted by ∩i∈I Ai, is the set {12, 24, 36, 48,...}. This means that the intersection contains all ______ of all sets Ai.

The union of sets includes elements that are in exactly one set.

Explain the meaning of the symbol '∃' in the context of sets?

What is the term used to describe the set of all mappings from A to B?

A correspondence is a mapping if and only if for every element of A, there is only one corresponding element in B.

What is the notation used to represent a mapping 'f' from A to B?

The element in B that corresponds to an element x in A, by the mapping f, is called the ______ of x.

Match the following terms with their corresponding definitions:

If (x, y) belongs to the graph G of a correspondence C from A to B, what does it signify?

In a correspondence, an element from the domain can have multiple corresponding elements from the codomain.

What term is used to describe the element of A in a correspondence (A, B, G)?

What is the notation used to represent the direct image of a subset X under a mapping f?

If a ∈ A and f(a) ∈ f(X), then a must be an element of X.

What is the direct image of the set X = {-2, -1, 1, 2} under the mapping f(x) = x²?

If X is an empty set, then the direct image of X under any mapping f is also an ______ set.

Which of the following statements is TRUE regarding the direct images of subsets under a mapping?

Match the following properties of mappings with their corresponding definitions:

In the context of a magma (E, >), an element e of E is an identity for > if and only if γe = δe = ______

According to the provided content, a magma can have multiple identity elements.

What is the condition for an element a in a magma (E, >) to be considered idempotent for >?

Which of the following statements accurately describes the concept of a left invertible element 'a' in a unitary magma (E, >) with identity element 'e'?

In the monoid (N, +), every element is invertible.

Match the following terms with their appropriate definitions:

Which of the following statements is true regarding the mapping f: A -> B and a family of subsets (Yi)i∈I of B?

For any subset X of A, f(X) is always a subset of f^(-1)(f(X))?

In Proposition 5.5, what is the implication of the statement '∀ X ∈ P(A), we have X ⊆ f^(-1)(f(X))'?

A mapping f: A -> B is ______ if and only if for any subset X of A, f^(-1)(f(X)) = X.

Match the following conditions with their corresponding property of a mapping:

Which of the following is NOT a condition equivalent to f being injective?

If f(X ∩ X') = f(X) ∩ f(X') for all X, X' ∈ P(A), then f is necessarily injective.

Describe the relationship between the surjectivity of a mapping f: A -> B and the image of the preimage of a subset Y of B.

Which of the following is NOT a characteristic of a mapping?

The correspondence defined in Example 1.1 is a mapping.

What is the name given to the mapping that maps each element of a set to itself?

The image or range of a mapping f is denoted by ______.

Match the following mappings with their descriptions:

Two mappings f and g are equal if they have the same domain and codomain.

What is the restriction of a mapping f to a subset X of its domain?

The composition of two mappings f: A → B and g: B → C is possible if:

Study Notes

Logic and Theory of Sets

Sets

Operations on Sets

Mappings/Functions

Subsets and Power Set

Methods of Proof

Studying That Suits You

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