Discrete Structures: Sequence, Relations, and Functions

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.