D421 Discrete Math: Functions and Relations - Set 4 - Part 3

Correct Answer: |A| = |B|

A bijection pairs elements one to one and onto, so the finite sets must have equal cardinality.

Correct Answer: A mapping defined on only part of the domain

A partial function assigns outputs to only some elements of the domain, leaving others undefined, unlike a total function.

Correct Answer: An equivalence relation

Parallelism (with each line parallel to itself) is reflexive, symmetric, and transitive, making it an equivalence relation.

Correct Answer: A partial order

Divisibility on positive integers is reflexive, antisymmetric, and transitive, so it is a partial order.

Correct Answer: The set of pairs (x, f(x))

A function's graph is the relation consisting of all ordered pairs (x, f(x)) for x in the domain.

Correct Answer: A function

If every vertical line crosses a graph at most once, each input has one output, so the graph represents a function.

Correct Answer: 25

A × A contains all ordered pairs of elements from A, giving 5 × 5 = 25 pairs.

Correct Answer: The pairs (a, a) for each element a

The identity relation consists of precisely the diagonal pairs (a, a), relating each element only to itself.

Correct Answer: Every function is a relation, but not every relation is a function

A function is a special relation where each input has exactly one output, so all functions are relations but not conversely.

Correct Answer: Permutation

A bijection from a set to itself rearranges its elements and is called a permutation of the set.