Lecture 4.1: Relations Basics CS 250, Discrete Structures, Fall 2014 Nitesh Saxena Adopted from previous lectures by Cinda Heeren

Course Admin Graded HW2 HW3 Due Will distribute today Covers “Induction and Recursion” (Chapter 5) Due 11am Nov 6 (Thursday) 7/2/2018

Course Admin Mid Term 2 Nov 6 (Thur) Covers “Induction and Recursion” (Chapter 5) Review Oct 30 (Thu) Study topics posted Sample practice exam will be given out soon 7/2/2018

Traveling Next Week At a conference (Nov 3-7): http://www.sigsac.org/ccs/CCS2014/ Mid-term proctored by TA No class next Tuesday (Nov 4) – prepare for exam Office hrs extended – 3-5pm Thursday. 7/2/2018

Outline Relation Definition and Examples Types of Relations Operations on Relations 7/2/2018

A x B = {<x,y> : xA and yB}. Relations Recall the definition of the Cartesian (Cross) Product: The Cartesian Product of sets A and B, A x B, is the set A x B = {<x,y> : xA and yB}. A relation is just any subset of the CP!! R  AxB Ex: A = students; B = courses. R = {(a,b) | student a is enrolled in class b} We often say: aRb if (a,b) belongs to R 7/2/2018

Relations vs. Functions Recall the definition of a function: f = {<a,b> : b = f(a) , aA and bB} Is every function a relation? Draw venn diagram of cross products, relations, functions Yes, a function is a special kind of relation. 7/2/2018

Properties of Relations Reflexivity: A relation R on AxA is reflexive if for all aA, (a,a) R. Symmetry: A relation R on AxA is symmetric if (a,b)  R implies (b,a)  R . 7/2/2018

Properties of Relations   7/2/2018

Properties of Relations - Techniques How can we check for the reflexive property? Draw a picture of the relation (called a “graph”). Vertex for every element of A Edge for every element of R Now, what’s R? {(1,1),(1,2),(1,3),(1,4),(2,2),(2,3),(2,4),(3,3),(3,4),(4,4)} 1 2 3 4 Loops must exist on EVERY vertex. 7/2/2018

Properties of Relations - Techniques How can we check for the symmetric property? Draw a picture of the relation (called a “graph”). Vertex for every element of A Edge for every element of R Now, what’s R? {(1,1),(1,2),(1,3),(1,4),(2,2),(2,3),(2,4),(3,3),(3,4),(4,4)} EVERY edge must have a return edge. 1 2 3 4 7/2/2018

Properties of Relations - Techniques How can we check for transitivity? Draw a picture of the relation (called a “graph”). Vertex for every element of A Edge for every element of R Now, what’s R? {(1,1),(1,2),(1,3),(1,4),(2,2),(2,3),(2,4),(3,3),(3,4),(4,4)} 1 2 3 4 A “short cut” must be present for EVERY path of length 2. 7/2/2018

Properties of Relations - Techniques How can we check for the anti-symmetric property? Draw a picture of the relation (called a “graph”). Vertex for every element of A Edge for every element of R Now, what’s R? {(1,1),(1,2),(1,3),(1,4),(2,2),(2,3),(2,4),(3,3),(3,4),(4,4)} No edge can have a return edge. 1 2 3 4

Properties of Relations - Examples Let R be a relation on People, R={(x,y): x and y have lived in the same country} 1 ? 1 2 ? 1 2 3 ? Is R transitive? No Is it symmetric? Yes Is it reflexive? Yes Is it anti-symmetric? No 7/2/2018

Properties of Relations - Examples Let R be a relation on positive integers, R={(x,y): 3|(x-y)} Suppose (x,y) and (y,z) are in R. Definition of “divides” Then we can write 3j = (x-y) and 3k = (y-z) Can we say 3m = (x-z)? Is (x,z) in R? Add prev eqn to get: 3j + 3k = (x-y) + (y-z) 3(j + k) = (x-z) Is R transitive? Yes 7/2/2018

Properties of Relations - Techniques Let R be a relation on positive integers, R={(x,y): 3|(x-y)} Is (x,x) in R, for all x? Definition of “divides” Does 3k = (x-x) for some k? Yes, for k=0. Is R transitive? Yes Is it reflexive? Yes 7/2/2018

Properties of Relations - Techniques Let R be a relation on positive integers, R={(x,y): 3|(x-y)} Suppose (x,y) is in R. Definition of “divides” Then 3j = (x-y) for some j. Does 3k = (y-x) for some k? Yes, for k=-j. Is R transitive? Yes Is it symmetric? Yes Is it reflexive? Yes 7/2/2018

Properties of Relations - Techniques Let R be a relation on positive integers, R={(x,y): 3|(x-y)} Suppose (x,y) is in R. Definition of “divides” Then 3j = (x-y) for some j. Does 3k = (y-x) for some k? Yes, for k=-j. Is R transitive? Yes Is it symmetric? Yes Is it reflexive? Yes Is it anti-symmetric? No 7/2/2018

More than one relation Suppose we have 2 relations, R1 and R2, and recall that relations are just sets! So we can take unions, intersections, complements, symmetric differences, etc. There are other things we can do as well… 7/2/2018

More than one relation SR = {(a,c):  bB, (a,b)  R, (b,c)  S} Let R be a relation from A to B (R  AxB), and let S be a relation from B to C (S  BxC). The composition of R and S is the relation from A to C (SR  AxC): SR = {(a,c):  bB, (a,b)  R, (b,c)  S} A B C 1 2 3 4 x y z s t u v R S SR = {(1,u),(1,v),(2,t),(3,t),(4,u)} 7/2/2018

R2 = R1R = {(1,1),(1,2),(1,3),(2,3),(3,3),(4,1), (4,2)} More than one relation Let R be a relation on A. Inductively define R1 = R Rn+1 = Rn  R A A A R R1 1 1 1 2 2 2 3 3 3 4 4 4 R2 = R1R = {(1,1),(1,2),(1,3),(2,3),(3,3),(4,1), (4,2)} 7/2/2018

R3 = R2R = {(1,1),(1,2),(1,3),(2,3),(3,3),(4,1),(4,2),(4,3)} More than one relation Let R be a relation on A. Inductively define R1 = R Rn+1 = Rn  R A A A R R2 1 1 1 2 2 2 3 3 3 … = R4 = R5 = R6… 4 4 4 R3 = R2R = {(1,1),(1,2),(1,3),(2,3),(3,3),(4,1),(4,2),(4,3)} 7/2/2018

Today’s Reading Rosen 9.1 7/2/2018