This chapter explores key concepts of relations and functions, including types of relations, properties of functions, and their compositions. Understanding these concepts is crucial for further studies in mathematics.
Relations and Functions – Formula & Equation Sheet
Essential formulas and equations from Mathematics Part - I, tailored for Class 12 in Mathematics.
This one-pager compiles key formulas and equations from the Relations and Functions chapter of Mathematics Part - I. Ideal for exam prep, quick reference, and solving time-bound numerical problems accurately.
Key concepts & formulas
Essential formulas, key terms, and important concepts for quick reference and revision.
Formulas
R = A × B
R represents a relation from set A to set B, defined as the Cartesian product of A and B. This establishes a framework for relating elements from two sets.
R = ∅
R is an empty relation when it contains no pairs, meaning no elements of A are related to elements of B. It is a subset of A × B.
R = A × A
R is a universal relation if every element in A relates to every element in A. This is the largest possible relation in terms of pairs.
a R b ⇔ (a, b) ∈ R
This notation denotes that element 'a' is related to element 'b' under relation R. It serves as a concise way to express relations.
(a, a) ∈ R for all a ∈ A
This condition defines a reflexive relation R on set A, indicating that every element is related to itself.
If (a₁, a₂) ∈ R, then (a₂, a₁) ∈ R
This statement defines a symmetric relation indicating that the order of elements in the pairs does not affect their relation.
If (a₁, a₂) ∈ R and (a₂, a₃) ∈ R, then (a₁, a₃) ∈ R
This defines a transitive relation, describing how relations can be chained together between elements.
f: X → Y
This notation defines a function f that maps elements from set X (domain) to set Y (co-domain).
f is injective if f(x₁) = f(x₂) ⇒ x₁ = x₂
This defines a one-to-one function (injective), where each input maps to a distinct output.
f is surjective if ∀ y ∈ Y, ∃ x ∈ X such that f(x) = y
A function is onto (surjective) when every element in the co-domain has a pre-image in the domain.
Equations
x R y ⇔ y ∈ f(x)
This notation defines a relation based on the output of a function f for an element x.
R(a) = {b ∈ B | (a, b) ∈ R}
This defines the range of an element a under the relation R, highlighting all elements b in set B related to a.
f(g(x))
This represents the composition of two functions, where the output of function g becomes the input for function f.
f(g(x)) = x
For an invertible function, this equality holds, signifying that the composition of a function and its inverse yields the identity element.
A = {1, 2, 3} → B = {4, 5}
This defines a mapping from set A to set B, demonstrating the fundamental concepts of functions.
g: X → Y, g(x) = ax^2 + bx + c
This defines a quadratic function, which is a specific type of function mapping inputs from domain X to outputs in co-domain Y.
f(x) + g(x)
This represents the sum of two functions, indicating how different functions can interact to produce new outputs.
R = {(x, y) | x is related to y}
This expresses a relation R as a set of ordered pairs, providing a way to visualize relationships.
f: X → Y and f is bijective
This indicates that the function f is both one-to-one and onto, ensuring that it has an inverse function.
f^-1: Y → X
This denotes the inverse function f^-1 which reverses the mapping of function f, applicable when f is bijective.
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