Harvard University, Math 101, Spring 2015 Binary relations and Functions 1 Binary Relations Intuitively, a binary relation is a rule to pair elements of a sets A to element of a set B. When two elements a 2 A is in a relation to an element b 2 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(A2). 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) = fx 2 A : 9 y 2 B xRyg Ran(R) = fy 2 B : 9 x 2 A : xRyg. 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 8x; y; z 2 A :(xRy and yRz) =) xRz: • R is reflexive iff 8x 2 A : x Rx • R is irreflexive iff 8x 2 A : :(xRx) • R is symmetric iff 8x; y 2 A : xRy =) yRx • R is asymmetric iff 8x; y 2 A : xRy =):(yRx): 1 • R is antisymmetric iff 8x; y 2 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: IdA = f(x; x): x 2 Ag: 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. 8 R is reflexive : 8x 2 A : xRx <> R is a relation of equivalence () R is transitive : 8x; y; z 2 A xRy ^ yRz =) xRz :>R is symmetric : 8x; y 2 A : xRy =) yRx A partition of a set A is by definition a union of subsets Ai that cover A but do not intersect each other: [ A = Ai; 8i; j; Ai \ Aj = ?: i Given a relation of equivalence, we denote by Cl(x) the class of equivalence of an element x: Cl(x) = fy 2 A : xRyg: Two elements have the same class if and only if they are in relation: Cl(x1) = Cl(x2) () x1Rx2: 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 anti- symmetric. 8 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 6= A We usually use " ≤ " for a relation of order. In that case we use \<'" for the corresponding strict order: x < y () (x ≤ y ^ x 6= y): A strict order is transitive,irreflexive and asymmetric: 8 R is irreflexive <> R is a relation of strict order () R is transitive :>R is asymmetric 2 • 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 8y 2 A : y comparable to x =) y ≤ x In other words, @y 2 A : x ≤ y • In a poset, an element x is called a minimal element if and only if 8y 2 A :(y comparable to x) =) x ≤ y or equivalently @y 2 A : y ≤ x • An element y is said to cover an element x if and only if (x ≤ y) ^ @z 2 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: 8 x 2 A9! y 2 B : xRy We recall that the notation 9! 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 • A function f : A ! B is injective if 8x1; x2 2 A : f(x1) = f(x2) =) x1 = x2: • A function f : A ! B is surjective if 8y 2 B 9x 2 A : f(x) = y: • A function is bijective if it is both injective and surjective: ( 8x 2 A 9!y 2 B : f(x) = y f : A ! B is bijective () 8y 2 A 9!x 2 A : 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 2 Im(A), the equation f(x) = y has one and only one solution. A function is surjective if for any element y 2 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) = fx 2 A : f(x) 2 Cg This is called the inverse image or the pre-image of C. Properties: −1 −1 −1 f (C1 [ C2) = f (C1) [ f (C2) −1 −1 −1 f (C1 \ C2) = f (C1) \ f (C2) −1 −1 −1 f (C1 − C2) = f (C1) − f (C2) −1 −1 f (C1) = f (C1) −1 −1 C1 ⊂ C2 =) f (C1) ⊂ f (C2) 7 Composition of functions 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) gL : B ! A if gL ◦ f = IdA: A function f : A ! B has a right-inverse ( or section) gR : B ! A if f ◦ gR = IdB: 4 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 gL(y) = x () y = f(x): The right-inverse of a surjective function is not uniquely defined, but it also satisfies the relation gR(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.