Set Theory Quiz

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: