Download presentation

Presentation is loading. Please wait.

1
1 Section 7.1 Relations and their properties

2
2 Binary relation A binary relation is a set of ordered pairs that expresses a relationship between elements of 2 sets Formal definition: –Let A and B be sets –A binary relation from A to B is a subset of AxB (Cartesian product)

3
3 Suppose a A and b B If (a,b) R, then aRb If (a,b) R, then aRb If aRb, we can state that a is related to b by R Denotation of binary relation R

4
4 Example 1 Let A = {0,1,2} and B = {a,b} Then {(0,a), (0,b), (1,a), (2,b)} is a relation from A to B We can state that, for instance, 0Ra and 1Rb We can represent relations graphically, as shown on the next slide

5
5 Example 1 A = {0,1,2} B = {a,b} R = {(0,a), (0,b), (1,a), (2,b)} 0a1b20a1b2 Rab0xx1x2xRab0xx1x2x

6
6 Functions as relations A function f from set A to set B assigns a unique element of B to each element of A The graph of f is the set of ordered pairs (a,b) such that b = f(a) The graph of f is a subset of AxB, so it is a relation from A to B

7
7 Functions as relations The graph of f has the property that every element of A is the first element of exactly one ordered pair of the graph If R is a relation from A to B such that every element is the first element of exactly one ordered pair of R, then a function can be defined with R as its graph

8
8 Not all relations are functions A relation can express a one-to-many relationship between elements of sets A and B, where an element of A may be related to several elements of B On the other hand, a function represents a relation in which exactly one element of B is related to each element of A

9
9 Relations on a set A relation on a set A is a relation from A to A; in other words, a subset of AxA Example: Let A = {1,2,3,4,5,6}; which ordered pairs are in the relation R={(a,b)|a divides b}? Solution: {(1,1) (1,2), (1,3), (1,4), (1,5), (1,6), (2,2), (2,4), (2,6), (3,3), (3,6), (4,4), (5,5), 6,6)}

10
10 Relations on the set of integers Relations on the set of integers are infinite relations Some examples include: R 1 = {(a,b) | a = b} R 2 = {(a,b) | a = 5b} R 3 = {(a,b) | a = b+2}

11
11 Finding the number of relations on a finite set A relation on a set A is a subset of AxA If A has n elements, AxA has n 2 elements A set with m elements has 2 m subsets Therefore, there are 2 n 2 relations on a set with n elements For set {a,b,c,d} there are 2 16, or 65,536 relations on the set

12
12 Properties of relations Reflexive: a relation R on set A is reflexive if (a,a) R for every element a A For example, for set A = {1,2,3} –if R = {(1,1), (1,2), (2,2), (3,1), (3,3)} then R is a reflexive relation –On the other hand, if R = {(1,1), (1,2), (2,3), (3,3)} then R is not a reflexive relation

13
13 Properties of relations Symmetric: a relation R on a set A is symmetric if (b,a) R whenever (a,b) R for all a,b A For set A = {a,b,c,d}: –if R = {(a,b), (b,a), (c,d), (d,c)} then R is symmetric –if R = {(a,b), (b,a), (c,d), (c,b)} then R is not symmetric

14
14 Properties of relations Antisymmetric: a relation R on a set A is antisymmetric if (a,b) R and (b,a) R only when a=b Note that symmetric and antisymmetric are not necessarily opposite; a relation can be both at the same time

15
15 Examples of symmetry and antisymmetry For A={1,2,3}: –R = {(1,1), (1,2), (2,1)} is symmetric but not antisymmetric –R = {(1,1), (1,2), (2,3)} is antisymmetric but not symmetric –R = {} is both symmetric and antisymmetric –R = {(1,2), (1,3), (2,3)} is antisymmetric

16
16 Properties of relations Transitive: A relation R on a set A is called transitive if, whenever (a,b) R and (b,c) R, then (a,c) R for a,b,c A For set A = {1, 2, 3, 4}: –R = {(1,3), (3,4), (1,2), (2,3), (2,4), (1,4)} is transitive –R = {(1,3), (3,4), (1,2), (2,4)} is not transitive

17
17 Example 2 Let A = set of integers and –R 1 = {(a,b) | a b} –R 2 = {(a,b) | a<b} –R 3 = {(a,b) | a=b or a=-b} –R 4 = {(a,b) | a=b} –R 5 = {(a,b) | a=b+1} –R 6 = {(a,b) | a+b 2} Which of these are reflexive, symmetric, antisymmetric, transitive?

18
18 Combining relations Since relations from A to B are subsets of AxB, relations from A to B can be combined any way 2 sets can be combined Let A={1,2,3} and B={1,2,3,4} and R 1 ={(1,1), (2,2), (3,3)}, R 2 ={(1,1),(1,2),(1,3),(1,4)} –R1 R2 = {(1,1), (1,2), (2,2), (1,3),(3,3), (1,4)} –R1 R2 = {(1,1)} –R1 - R2 = {(2,2), (3,3)} –R2 - R1 = {(1,2), (1,3), (1,4)}

19
19 Composition of relations Let R be a relation from A to B and S be a relation from B to C S R is the relation consisting of ordered pairs (a,c) where a A and c C and there exists an element b B such that (a,b) R and (b,c) S

Similar presentations

© 2024 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google