1 / 19

Section 7.1

Section 7.1. Relations and their properties. 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).

azana
Download Presentation

Section 7.1

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Section 7.1 Relations and their properties

  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. Denotation of binary relation R • 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

  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. Example 1 A = {0,1,2} B = {a,b} R = {(0,a), (0,b), (1,a), (2,b)} R a b 0 x x 1 x 2 x 0 a 1 b 2

  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. 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. 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. 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. Relations on the set of integers • Relations on the set of integers are infinite relations • Some examples include: R1 = {(a,b) | a = b} R2 = {(a,b) | a = 5b} R3 = {(a,b) | a = b+2}

  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 n2 elements • A set with m elements has 2m subsets • Therefore, there are 2n2 relations on a set with n elements • For set {a,b,c,d} there are 216, or 65,536 relations on the set

  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. 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. 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. 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. 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. Example 2 • Let A = set of integers and • R1 = {(a,b) | ab} • R2 = {(a,b) | a<b} • R3 = {(a,b) | a=b or a=-b} • R4 = {(a,b) | a=b} • R5 = {(a,b) | a=b+1} • R6 = {(a,b) | a+b2} • Which of these are reflexive, symmetric, antisymmetric, transitive?

  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 R1={(1,1), (2,2), (3,3)}, R2={(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. 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

More Related