10 Questions
What is the concept of relation used for?
Relating two objects or quantities with each other.
Define relation mathematically.
A relation R from a set A to a set B is a subset of the cartesian product A × B obtained by describing a relationship between the first element x and the second element y of the ordered pairs in A × B.
What is an empty relation?
If no element of A is related to any element of A.
What are the extreme relations in a set A?
The empty set φ and A × A.
What do students learn about in Chapter 1 of Relations and Functions for Class 12?
Different types of relations and functions, the composition of functions, etc., in detail.
What is the definition of a universal relation?
A relation R in a set A is a universal relation if R = A × A
Define an equivalence relation.
A relation R in a set A is an equivalence relation if R is reflexive, symmetric, and transitive
What is the requirement for a function to be considered invertible?
A function f : X → Y is invertible if there exists a function g : Y → X such that gof = IX and fog = IY
Explain the composition of functions.
The composition of functions f and g, denoted by gof, is defined as gof (x) = g(f (x)) for every x in A
What is the definition of a binary operation on a set A?
A binary operation ∗ on a set A is a function ∗ : A × A → A
Study Notes
Relations and Functions
- The concept of relation is used to describe a connection or association between elements of two sets.
Mathematical Definition of Relation
- A relation R from set A to set B is a subset of the Cartesian product A × B, where A × B is the set of all ordered pairs (a, b) such that a ∈ A and b ∈ B.
Empty Relation
- An empty relation is a relation that has no elements, denoted as R = ∅.
Extreme Relations
- In a set A, the extreme relations are:
- The universal relation, denoted as A × A, which is the set of all ordered pairs (a, b) where a, b ∈ A.
- The empty relation, denoted as ∅, which is the set with no ordered pairs.
Chapter 1 of Relations and Functions for Class 12
- Students learn about the basics of relations and functions, including the definition of relations, types of relations, and operations on relations.
Universal Relation
- A universal relation is a relation that contains all possible ordered pairs of a set A, denoted as A × A.
Equivalence Relation
- An equivalence relation is a relation that is reflexive, symmetric, and transitive, denoting a "sameness" or "equivalence" relation between elements.
Invertible Function
- A function is considered invertible if it is both one-to-one (injective) and onto (surjective).
Composition of Functions
- The composition of functions f and g, denoted as (f ∘ g), is a function that combines the output of g as the input of f, resulting in a new function.
Binary Operation
- A binary operation on a set A is a function that takes two elements of A as input and produces another element of A as output, denoted as ∘: A × A → A.
Test your knowledge of relations and functions in Class 12 Mathematics Chapter 1. This quiz covers the revision of general notation, domain, codomain, range, real-valued functions, composition of functions, and more.
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