RELATIONS AND FUNCTIONS
Frequently Asked Questions
An ordered pair is a pair of elements written as \((a, b)\), where the order matters. Two ordered pairs are equal if and only if their corresponding elements are equal.
The Cartesian product of sets \(A\) and \(B\), denoted \(A \times B\), is the set of all ordered pairs \((a, b)\) where \(a \in A\) and \(b \in B\).
If set \(A\) has \(m\) elements and set \(B\) has \(n\) elements, then \(A \times B\) has \(m \times n\) elements.
A relation from set \(A\) to set \(B\) is any subset of the Cartesian product \(A \times B\).
The domain is the set of all first elements of the ordered pairs belonging to the relation.
The range is the set of all second elements of the ordered pairs of a relation.
The codomain is the set from which the second elements of ordered pairs are taken, regardless of whether all elements appear in the relation or not.
A relation that contains no ordered pair is called an empty relation.
A relation that contains all possible ordered pairs of a Cartesian product is called a universal relation.
An identity relation on a set \(A\) consists of all ordered pairs \((a, a)\) for every \(a \in A\).
A relation is reflexive if every element of the set is related to itself, i.e., \((a, a)\) belongs to the relation for all \(a\).
A relation is symmetric if whenever \((a, b)\) belongs to the relation, \((b, a)\) also belongs to it.
A relation is transitive if whenever \((a, b)\) and \((b, c)\) belong to the relation, then \((a, c)\) must also belong to it.
A relation that is reflexive, symmetric, and transitive is called an equivalence relation.
An equivalence class is the set of all elements related to a given element under an equivalence relation.
A function is a special type of relation in which every element of the domain is associated with exactly one element of the codomain.
In a relation, an element of the domain may have multiple images or none, whereas in a function each domain element has exactly one image.
A function is one-one if distinct elements of the domain have distinct images in the codomain.
A function is onto if every element of the codomain has at least one pre-image in the domain.
A function that is both one-one and onto is called a bijective function.
Bijective functions allow the definition of inverse functions and establish a perfect one-to-one correspondence between two sets.
A real-valued function is a function whose domain and codomain are subsets of the set of real numbers.
A function is represented graphically by plotting ordered pairs \((x, f(x))\) on the Cartesian plane.
The vertical line test states that a graph represents a function if and only if no vertical line intersects the graph at more than one point.
Relations and Functions form the foundation for calculus, coordinate geometry, matrices, and real analysis studied in higher classes.
They are used to model dependencies such as temperature variation with time, cost with quantity, population growth, and physical laws.
Definition-based, reasoning-based, relation classification, domain-range identification, and function-type identification questions are common.
Focus on definitions, properties, standard examples, and clear logical explanations with proper mathematical notation.
Yes, it builds the conceptual base required for functions, graphs, and mappings used extensively in higher-level problems.
It enhances logical reasoning, abstract thinking, precise mathematical communication, and analytical problem-solving skills.
The image of an element \(x\) under a function \(f\) is the value \(f(x)\) in the codomain corresponding to \(x\).
A pre-image of an element \(y\) in the codomain is an element \(x\) in the domain such that \(f(x) = y\).
Yes, this occurs in many-one functions where distinct domain elements map to the same codomain element.
No, assigning more than one value to a single domain element violates the definition of a function.
If set \(A\) has \(m\) elements and set \(B\) has \(n\) elements, the total number of relations is \(2^{mn}\).
If set \(A\) has \(m\) elements and set \(B\) has \(n\) elements, the total number of functions is \(n^m\).
A transformation describes how a function maps elements from one set to another according to a specific rule or operation.
Arrow diagrams visually represent relations and functions, helping to identify whether a relation satisfies the conditions of a function.
It is a table listing elements of the domain alongside their corresponding images, useful for clarity and verification.
Confusing domain with codomain, assuming all relations are functions, and misidentifying one-one and onto properties are common errors.
The domain determines where the function is defined; incorrect domain selection can change the nature and validity of a function.
Restriction of domain involves limiting the set of input values to ensure the function is well-defined or satisfies required properties.
Concepts of functions, domain, and range are essential for understanding limits, continuity, and derivatives.
While not mandatory, graphs provide intuitive understanding and help verify whether a relation represents a function.
They help classify elements into distinct groups called equivalence classes based on shared properties.
Relations can be represented using matrices, and functions describe mappings fundamental to linear algebra.
Questions testing logical consistency, property verification, and classification of relations and functions are common.
Answers should include clear definitions, correct notation, logical steps, and proper justification wherever required.
It is largely theory-based but requires strong conceptual understanding to solve reasoning and classification problems.
It establishes the foundational language and structure for expressing mathematical relationships used throughout higher mathematics.
