1. Fall 2015 COMP 2300 Discrete Structures for Computation Donghyun (David) Kim Department of Mathematics and Physics North Carolina Central University 1 Chapter 8.2 Reflexivity, Symmetry, and Transitivity

2. Definitions Informally, a relation R is Reflexive: Each element is related to itself. 2 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

3. Definitions – cont’ Informally, a relation R is Symmetric: If any one element is related to any other element, then the second element is related to the first. 3 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

4. Definitions – cont’ Informally, a relation R is Transitive: If any one element is related to a second and that second element is related to a third, then the first element is related to the third. 4 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

5. Definitions – cont’ Let R be a relation on a set A. R is reflexive if, and only if, for all R is symmetric if, and only if, for all, if then. R is transitive if, and only if, for all x, y, z in A, if and, then Or, equivalently, R is reflexive for all R is symmetric for all if then R is transitive for all, if and, then 5 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

6. Definitions – cont’ Let R be a relation on a set A. R is NOT reflexive There is an element x in A such that R is NOT symmetric There are elements x and y in A such that but. R is NOT transitive There are elements x, y, and z in A such that and but. 6 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

7. Properties of Relations on Finite Sets Let A = {0, 1, 2, 3} and define relations R, S, and T on A as follows: a.Is R reflexive? Symmetric? Transitive? b.Is S reflexive? Symmetric? Transitive? c.Is T reflexive? Symmetric? Transitive? Use the directed graph of R, S, and T to answer the questions. 7 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

8. Properties of Relations on Finite Sets – cont’ Let A = {0, 1, 2, 3} and define relations R, S, and T on A as follows: a.Is R reflexive? Symmetric? Transitive? 8 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

9. Properties of Relations on Finite Sets – cont’ Let A = {0, 1, 2, 3} and define relations R, S, and T on A as follows: a.Is R reflexive? Symmetric? Transitive? 9 Reflexive, Symmetric, but not Transitive. Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

10. Properties of Relations on Finite Sets – cont’ Let A = {0, 1, 2, 3} and define relations R, S, and T on A as follows: b.Is S reflexive? Symmetric? Transitive? 10 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

11. Properties of Relations on Finite Sets – cont’ Let A = {0, 1, 2, 3} and define relations R, S, and T on A as follows: b.Is S reflexive? Symmetric? Transitive? 11 Not Reflexive, not Symmetric, but Transitive. Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

12. Properties of Relations on Finite Sets – cont’ Let A = {0, 1, 2, 3} and define relations R, S, and T on A as follows: c.Is T reflexive? Symmetric? Transitive? 12 Not Reflexive, not Symmetric, but Transitive. Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

13. Properties of Relations on Infinite Sets Suppose a relation R is defined on an infinite set A. To prove the relation is reflexive, symmetric, or transitive, first write down what is to be proved. For instance, for symmetry you need to prove that Then, use the definitions of A and R to rewrite the statement for the particular case in question. For instance, for the “equality” relation on the set of real numbers, the rewritten statement is Sometimes the truth of the rewritten statement will be immediately obvious. 13 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

14. Properties of Equality Define a relation R on R as follows: For all real numbers x and y, a.Is R reflexive? b.Is R symmetric? c.IS R transitive? 14 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

15. Properties of Equality – cont’ Define a relation R on R as follows: For all real numbers x and y, a.Is R reflexive? This is true since for all b.Is R symmetric? This is true since for all c.IS R transitive? This is true since for all 15 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

16. Properties of Less Than Define a relation R on R as follows: For all real numbers x and y, a.Is R reflexive? b.Is R symmetric? c.IS R transitive? 16 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

17. Properties of Less Than Define a relation R on R as follows: For all real numbers x and y, a.Is R reflexive? This is NOT true since for all For example, b.Is R symmetric? This is NOT true since for all c.IS R transitive? This is true since for all 17 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

18. The Transitive Closure of a Relation Let A be a set and R a relation on A. The transitive closure of R is the relation on A that satisfies the following three properties: 1. is transitive. 2. 3.If S is any other transitive relation that contains R, then 18 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

19. The Transitive Closure of a Relation – cont’ Q) Let A ={0,1,2,3} and consider the relation R defined on A as follows: Find the transitive closure of R. A) Every ordered pair in R is in, so Thus the directed graph of R contains the arrows shown below. 19 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

20. The Transitive Closure of a Relation – cont’ 20 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University