Relations, Functions, Binary Relations (Chapter 1, Sections 1.2, 1.3)

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1 Relations, Functions, Binary Relations (Chapter 1, Sections 1.2, 1.3) CmSc 365 Theory of Computation 1. Relations Definition: Let A and B be two sets. A relation R from A to B is any set of ordered pairs (x,y), x є A, y є B. Thus R A x B. The notation xry is used to denote that two elements x and y are in relation R Definition: Let R be a relation from A to B. The inverse relation R -1 of R is defined as: R -1 = {(b,a) : (a,b) є R} Thus if xry, then yr -1 x. If x and y are in relation R, then y and x are in the inverse relation R Functions A function from a set A to a set B is a relation R with the property: Each element of A is a member of only one ordered pair of the relation. Notation: Let f be a function from A to B. We write f : A B a є A, f(a) = b, b є B 2.1. Definitions: Let f be a function from A to B. 1. Domain of f: the set A 2. Range of f: {b: b є B and there is an a є A, f(a) = b} 3. Image of a under f: f(a) 2.2. Functions with more arguments Let A = A1 x A2, and f be a function from A to B We write: f(a1,a2) = b If A = A1 x A2 x x An, we write f(a1,a2,,an) = b a1, a2,..,an: arguments of f b : value of f 1

2 2.3. Functions of special interest a. one-to-one distinct elements have distinct images if a1 a2, then f(a1) f(a2) b. onto Every element in B is an image of some element in A c. bijection f is bijection iff f is a one-to-one function and f is a onto function If f is a bijection, f -1 is a function, also a bijection. Otherwise f -1 fails to be a function. Properties: f -1 (f(a)) = a, f(f -1 (b)) = b, a є A, b є B, Examples: Relation, not a function: The relation R1 = {(a,1),(a,2),(b,3),(c,3)} is not a function, because the element a has two images. The relation R2 = {(a,1),(b,2)} is not a function, because the element c has no image. Function, not one-to-one, not onto: The relation R = {(a,1),(b,1),(c,3)} is a function, not one-to-one, not onto Function, one-to-one, not onto Let A = {a,b,c}, B = {1,2,3,4,5} The relation R = {(a,1),(b,2),(c,5)} is a function, one-to-one, not onto 2

3 Function, onto, not one-to-one Let A = {a,b,c,d}, B = {1,2,3} The relation R = {(a,1),(b,1),(c,2),(d,3)} is a function: onto, not one-to-one Function, bijection: The relation R = {(a,1),(b,2),(c,3)} is a function, bijection. Here the inverse relation is a function too. 3. Binary Relations Binary relation: a relation between two sets: arb, a є A, b є B, Binary relations on a set and itself: arb, a є A, b є A Binary relations on a set can be represented by a directed graph: A: {a,b,c,d,e,f} R: {(a,b),(a,d),(b,a),(b,b),(b,f),(c,d)} Definition: Let Q and R be two binary relations. The composition Q R (or just QR) is defined as: Q R = {(a,b): for some c, (a,c) є Q and (c,b) є R}. Composition of functions: Let f : A B, g : B C be two functions. The composition h = f g is a function from A to C such that h(a) = g(f(a)). Example: Let f (x) = x +1, g(x) = x 2. The composition h(x) = f (x) g(x) = g(f(x))= (x+1) 2 The composition p(x) = g(x) f(x) = f(g(x)) = x

4 4. Properties of Binary relations Reflexive relations: Irreflexive: neither: a є A, (a,a) є R a є A, (a,a) R a such that (a,a) є R, b such that (b,b) R Symmetric relations: Antisymmetric neither: (a,b) є R, a b, (a,b) є R, a b, (a,b) є R, a b, (b,a) є R (b,a) R such that (b,a) є R, (c,d) є R, c d, such that (d,c) R Transitive relations: Not transitive: If (a,b) є R and (b,c) є R, There are a, b, c such that then (a,c) є R (a,b) and (b,c) are in R, however (a,c) is not in R Relations of Equivalence: R is a relation of equivalence iff the relation is reflexive, symmetric and transitive A relation of equivalence specifies a partition of the set. Each element in the partition is called a class of equivalence. Examples: R1 = "having same mother" defines a partition on the set of all persons R2 = have same first letter defines a partition on the set of English words Partial orders: R is a partial order relation iff the relation is anti-symmetric and transitive. If the relation is also reflexive, the partial order is called weak partial order. If the relation is irreflexive, we have strict partial order Example: R "being a subset of a set" defines a weak partial order on power sets. A = {a,b,c} 4

5 2 A = {, {a}, {b}, {c}, {a,b}, {a,c}, {b,c}, {a,b,c} } The relation < defines strict partial order on the numbers Minimal element: a, such that for no b (b,a) є R. In the above example is the minimal element. Maximal element: b, such that for no a (b,a) є R In the above example {a, b, c} is the maximal element If the set is finite: at least one minimal and one maximal element Infinite sets the minimal and maximal elements are not guaranteed. Total order: a partial order relation that is defined for any two elements in the set. Example: R: "being less or equal to" defines a total order on the set of real numbers. Total order: not more than one minimal(maximal) element. 5