Binary relations and Functions

Transcription

1 1 Binary Relations Harvard University, Math 101, Spring 2015 Binary relations and Functions Intuitively, a binary relation is a rule to pair elements of a sets A to element of a set B. When two elements a A is in a relation to an element b B we write a R b. Since the order is relevant, we can completely characterize a relation R by the set of ordered pairs (a, b) such that a R b. This motivates the following formal definition: Definition A binary relation between two sets A and B is a subset of the Cartesian product A B. In other words, a binary relation is an element of P(A B). A binary relation on A is a subset of P(A 2 ). It is useful to introduce the notions of domain and range of a binary relation R from a set A to a set B: Dom(R) = {x A : y B xry} Ran(R) = {y B : x A : xry}. 2 Properties of a relation on a set Given a binary relation R on a set A, we have the following definitions: R is transitive iff x, y, z A : (xry and yrz) = xrz. R is reflexive iff x A : x Rx R is irreflexive iff x A : (xrx) R is symmetric iff x, y A : xry = yrx R is asymmetric iff x, y A : xry = (yrx). 1

2 R is antisymmetric iff x, y A : (xry and yrx) = x = y. In a given set A, we can always define one special relation called the identity relation. It is defined by the diagonal set: Id A = {(x, x) : x A}. 3 Partitions and Relation of equivalences A binary relation on a set A is said to be a relation of equivalence if it is reflexive, transitive and symmetric. R is reflexive : x A : xrx R is a relation of equivalence R is transitive : x, y, z A xry yrz = xrz R is symmetric : x, y A : xry = yrx A partition of a set A is by definition a union of subsets A i that cover A but do not intersect each other: A = A i, i, j, A i A j =. i Given a relation of equivalence, we denote by Cl(x) the class of equivalence of an element x: Cl(x) = {y A : xry}. Two elements have the same class if and only if they are in relation: Cl(x 1 ) = Cl(x 2 ) x 1 Rx 2. This is a direct consequence of transitivity and symmetry. We can show (Pset 4) that given a relation of equivalence on a set A, its classes of equivalence form a partition of the set A. 4 Total and partial order A binary relation on a set A is said to be a relation of order if it is reflexive, transitive and antisymmetric. R is reflexive R is a relation of order R is transitive R is antisymmetric If the domain of the relation is the full set A, the order relation is said to be a total order, otherwise it is called a partial order: { total order if and only if dom R = A A relation of order R is said to be a partial order if dom R A We usually use for a relation of order. In that case we use < for the corresponding strict order: x < y (x y x y). A strict order is transitive,irreflexive and asymmetric: R is irreflexive R is a relation of strict order R is transitive R is asymmetric 2

3 A set endowed with a partial order is sometimes called a partially ordered set or more briefly a poset. In a poset, two elements are said to be comparable if and only if x y or y x. From that point of view, a total order is an order in which all elements are comparable. A subset of a poset containing elements that are all comparable to each other is called a chain. In a poset, a subset such that any two distinct elements are not comparable to each other is called an anti-chain. In a poset, an element x is said to be a maximal element if and only if y A : y comparable to x = y x In other words, y A : x y In a poset, an element x is called a minimal element if and only if y A : (y comparable to x) = x y or equivalently y A : y x An element y is said to cover an element x if and only if (x y) z A : x < z < y. In other words, the interval (x, y) is empty. We also say that x is covered by y, this is denoted by x <: y. 5 Functions A function is a binary relation between two sets A and B such that any element of A is in a relation with one and only one element of B: x A! y B : xry We recall that the notation! means there exists one and only one. This definition of a function implies that the domain of the relation R is all of A and that for any element y of the range of the relation, the equation xry has one and only one solution. Notation: If xry and R is a function, we write y = R(x) since y is uniquely defined by x. Then y is called the image of x. We also write f : A B for a function from a set A to a set B. In the traditional definition of a function, A is the domain of the function: any element of A admits one and only one image in B. B is sometimes called the co-domain or target space of the function. 3

4 A function f : A B is injective if x 1, x 2 A : f(x 1 ) = f(x 2 ) = x 1 = x 2. A function f : A B is surjective if y B x A : f(x) = y. A function is bijective if it is both injective and surjective: { x A!y B : f : A B is bijective y A!x A : f(x) = y f(x) = y Remark A function f : A B is surjective if and only if Im(f) = B. A function is injective if for any element y Im(A), the equation f(x) = y has one and only one solution. A function is surjective if for any element y B, the equation f(x) = y has at least one solution. 6 Pre-image Given a subset C B, we denote by f 1 (C) the set of elements of A that are mapped to an element of C: f 1 (C) = {x A : f(x) C} This is called the inverse image or the pre-image of C. Properties: 7 Composition of functions f 1 (C 1 C 2 ) = f 1 (C 1 ) f 1 (C 2 ) f 1 (C 1 C 2 ) = f 1 (C 1 ) f 1 (C 2 ) f 1 (C 1 C 2 ) = f 1 (C 1 ) f 1 (C 2 ) f 1 (C 1 ) = f 1 (C 1 ) C 1 C 2 = f 1 (C 1 ) f 1 (C 2 ) We denote the composition of two functions f : A B and g : B C by g f. By definition, we have (f g)(x) = f[g(x)]. The composition of functions is not commutative but it is associative. 8 Inverse function A function f : A B has a left-inverse ( a retraction) g L : B A if g L f = Id A. A function f : A B has a right-inverse ( or section) g R : B A if f g R = Id B. 4

5 A left-inverse is also called a retraction while a right-inverse is also called a section. The left-inverse of an injective function is uniquely defined by the relation g L (y) = x y = f(x). The right-inverse of a surjective function is not uniquely defined, but it also satisfies the relation g R (y) = x y = f(x). Theorem (Pset 4) A function is injective if and only if it has a left-inverse. A function is surjective if and only if it has a right-inverse. Theorem Given an injective map f : A B, we can always define a bijection A Im(B) which is uniquely defined by the restriction of any left-inverse image of f restricted to Im(B). 5