Discrete Structures: Sequence, Relations, and Functions

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Questions and Answers

Which of the following best defines a sequence?

A collection of objects with repetitions allowed and order does not matter

A collection of objects without repetitions and order does not matter

A collection of objects with repetitions allowed and order matters (correct)

A collection of objects without repetitions and order matters

How can a sequence be logically represented in a finite sequence?

General form for arithmetic sequence

Matrix or Mapping

Roster form (correct)

Data structures: Array

What is the purpose of a Cartesian Product in the context of sequences?

It defines a sequence visually

It represents a logical construction of a sequence

It computes all possible ways to take things from multiple sets (correct)

It represents the programming representation of a sequence

In the programming representation of a sequence, what is the general form for an arithmetic sequence?

(General form for arithmetic sequence) Signup and view all the answers

Which of the following is NOT a valid mathematical representation of a sequence?

(General form for logical sequence) Signup and view all the answers

What is the formal definition of a sequence?

A function from natural numbers to elements at each position Signup and view all the answers

How is a sequence represented in the statement form?

Sequence of natural number in ascending form Signup and view all the answers

What is the visual design representation for sequences in software?

Data structures: Array (Table or Matrix) Signup and view all the answers

Which of the following does NOT represent a valid general form for sequence?

Sequence of natural number in descending form Signup and view all the answers

What is the construction purpose of a Cartesian Product in sequences?

All possible ways to take things from multiple sets Signup and view all the answers

In the context of the given text, what is a binary relation from set A to set B?

A subset of the Cartesian product of set A and set B Signup and view all the answers

Which property must a relation satisfy in order to be considered an equivalence relation?

Reflexive, symmetric, and transitive Signup and view all the answers

What does an equivalence relation on a set mean?

It classifies elements in the set into equivalent classes Signup and view all the answers

What is the cardinality of the Cartesian product of two sets with m and n elements, respectively?

$mn$ Signup and view all the answers

Which representation does not hold the commutative property in the context of relations?

Matrices Signup and view all the answers

What does it mean for one element to be related to another element in a relation?

There exists a directed edge between the two elements Signup and view all the answers

Which of the following is an example of an equivalence relation?

$x = y$ Signup and view all the answers

What is the cardinality of all possible relations from set A to set B?

$2^{mn}$ Signup and view all the answers

In terms of binary relations, what does it mean for two elements to be related?

They are connected by an edge Signup and view all the answers

Study Notes

Sequences

A sequence is a logical representation of a finite or infinite series of elements, often denoted as {a_n} or (a_n), where 'a' represents the sequence and 'n' represents the term number.
A sequence can be logically represented in a finite sequence by listing its elements, separated by commas, and enclosed in braces.

Cartesian Product

The purpose of a Cartesian Product in the context of sequences is to construct a new set from two given sets, containing all possible ordered pairs of elements from the original sets.
The construction purpose of a Cartesian Product in sequences is to represent the set of all possible combinations of elements from two sets.

Mathematical Representation of Sequences

The general form for an arithmetic sequence in programming representation is a_n = a_1 + (n-1)d, where 'a_n' is the nth term, 'a_1' is the first term, 'n' is the term number, and 'd' is the common difference.
A valid mathematical representation of a sequence is {a_n} or (a_n), where 'a' represents the sequence and 'n' represents the term number.
The formal definition of a sequence is a function from the set of natural numbers to a set, where each natural number corresponds to a specific term in the sequence.

Representation of Sequences

In statement form, a sequence is represented as a collection of statements, with each statement corresponding to a specific term in the sequence.
The visual design representation for sequences in software is often a flowchart or a diagram showing the sequence of steps or elements.

Relations

A binary relation from set A to set B is a subset of the Cartesian Product of A and B, where each element of the subset is an ordered pair (a, b) such that 'a' is from set A and 'b' is from set B.
An equivalence relation on a set satisfies the reflexive (every element is related to itself), symmetric (if 'a' is related to 'b', then 'b' is related to 'a'), and transitive (if 'a' is related to 'b' and 'b' is related to 'c', then 'a' is related to 'c') properties.
An equivalence relation on a set means that the relation satisfies the reflexive, symmetric, and transitive properties, and partitions the set into distinct classes or equivalence classes.

Cardinality of Cartesian Product and Relations

The cardinality of the Cartesian Product of two sets with 'm' and 'n' elements, respectively, is m × n.
The cardinality of all possible relations from set A to set B is 2^(|A|×|B|), where '|A|' and '|B|' represent the cardinality of sets A and B, respectively.

Properties of Relations

The representation that does not hold the commutative property in the context of relations is the Cartesian Product, as the order of the sets matters.
For one element to be related to another element in a relation means that there exists a relationship between the two elements, represented as an ordered pair in the relation.
An example of an equivalence relation is the "is equal to" relation (=) on the set of real numbers.
In terms of binary relations, for two elements to be related means that they satisfy the relation, and the relation can be represented as a subset of the Cartesian Product of the sets.

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Description

Test your knowledge of sequences, relations, and functions in discrete mathematics with this quiz. Explore the concepts of sequences as enumerated collections with allowed repetitions and ordered elements, defined formally as a function from natural numbers to the elements at each position.