Tiêu đề Chapter 1 Relations and Functions.pdf ✅

MATHEMATICS Chapter 1: Relation and Function
(1) RELATIONS AND FUNCTIONS 01 RELATIONS AND FUNCTIONS Top Concepts in Relations 1. Introduction to Relation and no. of relations ● A relation R between two non-empty sets A and B is a subset of their Cartesian product A × B. ● If A = B, then the relation R on A is a subset of A × A. ● The total number of relations from a set consisting of m elements to a set consisting of n elements is 2mn . ● If (a, b) belongs to R, then a is related to b and is written as ‘a R b’. If (a, b) does not belong to R, then a is not related to b and it is written as 2. Co-domain and Range of a Relation Let R be a relation from A to B. Then the ‘domain of and the ‘range of Co- domain is either set B or any of its superset or subset containing range of R. 3. Types of Relations A relation R in a set A is called an empty relation if no element of A is related to any element of A, i.e., A relation R in a set A is called a universal relation if each element of A is related to every element of A, i.e., R = A × A. 4. A relation R on a set A is called: a. Reflexive, if (a, a) ∈ R for every a ∈ A. b. Symmetric, if (a1, a2) ∈ R implies that (a2, a1) ∈ R for all a1, a2 ∈ A. c. Transitive, if (a1, a2) ∈ R and (a2, a3) ∈ R implies that (a1, a3) ∈ R for all a1, a2, a3 ∈ A. 5. Equivalence Relation ● A relation R in a set A is said to be an equivalence relation if R is reflexive, symmetric and transitive.
(2) RELATIONS AND FUNCTIONS 01 ● An empty relation R on a non-empty set X (i.e., ‘a R b’ is never true) is not an equivalence relation, because although it is vacuously symmetric and transitive, but it is not reflexive (except when X is also empty). 6. Given an arbitrary equivalence relation R in a set X, R divides X into mutually disjoint subsets Si called partitions or subdivisions of X provided: a. All elements of S, are related to each other for all i. b. No element of Si is related to any element of St if i ≠ j. c. The subsets St are called equivalence classes. 7. Union, Intersection and Inverse of Equivalence Relations a. If R and S are two equivalence relations on a set A, R ∩ S is also an equivalence relation on A. b. The union of two equivalence relations on a set is not necessarily an equivalence relation on the set. c. The inverse of an equivalence relation is an equivalence relation. Top Concepts in Functions 1. Introduction to functions A function from a non-empty set A to another non-empty set B is a correspondence or a rule which associates every element of A to a unique element of B written as f : A → B such that f(x) = y for all x ∈ A, y ∈ B. All functions are relations, but the converse is not true. 2. Domain, Co-domain and Range of a Function ● If f : A → B is a function, then set A is the domain, set B is the co-domain and set {f(x) : x ∈ A) is the range of f. ● The range is a subset of the co-domain. ● A function can also be regarded as a machine which gives a unique output in set B corresponding to each input from set A.
(3) RELATIONS AND FUNCTIONS 01 ● If A and B are two sets having m and n elements, respectively, then the total number of functions from A to B is nm. 3. Real Function ● A function f : A → B is called a real-valued function if B is a subset of R. ● If A and B both are subsets of R, then 'f' is called a real function. ● While describing real functions using mathematical formula, x (the input) is the independent variable and y (the output) is the dependent variable. ● The graph of a real function ‘f’ consists of points whose co-ordinates (x, y) satisfy y = f(x), for all x ∈ Domain(f). 4. Vertical line test A curve in a plane represents the graph of a real function if and only if no vertical line intersects it more than once. 5. One-one Function ● A function f : A → B is one-to-one if for all x, y ∈ A, f(x) = f(y) ⇒ x = y or x ≠ y ⇒ f(x) ≠ f(y). ● A one-one function is known as an injection or injective function. Otherwise, f is called many-one. 6. Onto Function ● A function f : A → B is an onto function, if for each b ∈ B, there is at least one a ∈ A such that f(a) = b, i.e., if every element in B is the image of some element in A, then f is an onto or surjective function. ● For an onto function, range = co-domain. ● A function which is both one-one and onto is called a bijective function or a bijection. ● A one-one function defined from a finite set to itself is always onto, but if the set is infinite, then it is not the case. 7. Let A and B be two finite sets and f : A → B be a function. ● If f is an injection, then n (A) ≤ n (B) . ● If f is a surjection, then n(A) ≥ n(B). ● If f is a bijection, then n(A) = n(B) . 8. If A and B are two non-empty finite sets containing m and n elements, respectively, then