Relations MATH Relations. Benjamin V.C. Collins, James A. Swenson MATH 2730

Transcription

1 MATH 2730 Benjamin V.C. Collins James A. Swenson

2 among integers equals a = b is true for some pairs (a, b) Z Z, but not for all pairs. is less than a < b is true for some pairs (a, b) Z Z, but not for all pairs. divides a b is true for some pairs (a, b) Z Z, but not for all pairs.

3 A relation is a set of ordered pairs. Given sets A and B, a relation from A to B is a subset of A B. Given a set A, a relation on A is a subset of A A. How does this definition correspond to the examples? equals is less than divides

4 equals is a relation on Z We see equals as a relation on Z, which means a subset of Z Z: {..., ( 2, 2), ( 1, 1), (0, 0), (1, 1), (2, 2),... }

5 is less than is a relation on Z We see is less than as a relation on Z: {(a, b) Z Z : a < b}

6 divides is a relation on Z We see divides as a relation on Z: {(a, b) Z Z : a b}

7 There are relations in every A B Given A = {1, 2, 3, 4} and B = {5, 6, 7}, is a relation from A to B: = {(1, 5), (1, 6), (3, 7), (4, 6), (4, 7)}

8 Many familiar things are relations is a relation from Z to the power set 2 Z. (Or you could replace Z by any other set.) One element of is (4, {2, 4, 6}).

9 Infix notation We often write arb as a shorthand for (a, b) R, and a R b when (a, b) R. Familiar examples a = b; a < b; a b 4 {2, 4, 6}

10 Infix notation We often write arb as a shorthand for (a, b) R, and a R b when (a, b) R. Another example Given A = {1, 2, 3, 4} and B = {5, 6, 7}, is a relation from A to B: = {(1, 5), (1, 6), (3, 7), (4, 6), (4, 7)} We say (for example) that 3 7, but 2 5.

11 Defining our own relation Example Let s define a relation on 2 Z called. Given sets U and V, we say U V provided U V. For example, {1, 3, 5} {3, 4, 5}. Equivalently, ({1, 3, 5}, {3, 4, 5}). On the other hand, Z.

12 The inverse of a relation Let R be a relation from A to B. The inverse of R is R 1 = {(b, a) B A : (a, b) R}. Example If = {(1, 5), (1, 6), (3, 7), (4, 6), (4, 7)}, then 1 = {(5, 1), (6, 1), (7, 3), (6, 4), (7, 4)}.

13 The inverse of a relation Let R be a relation from A to B. The inverse of R is R 1 = {(b, a) B A : (a, b) R}. Puzzles What are the inverses of the following relations? = < (Recall: U V provided U V.)

14 Reflexive and irreflexive Let A be any set, and let R be a relation on A (so R A A). The relation R is reflexive provided x A, xrx. The relation R is irreflexive provided x A, x R x. Can you visualize these properties in terms of the graph of R? Most relations are neither reflexive nor irreflexive!

15 Symmetric and antisymmetric R is symmetric provided x A, y A, (xry) (yrx). R is antisymmetric provided: x A, y A, (xry) (yrx) x = y. Can you visualize these properties in terms of the graph of R?

16 Symmetric and antisymmetric R is symmetric provided x A, y A, (xry) (yrx). R is antisymmetric provided: x A, y A, (xry) (yrx) x = y. Can you visualize these properties in terms of the graph of R? Most relations are neither symmetric nor antisymmetric!

17 Symmetric and antisymmetric R is symmetric provided x A, y A, (xry) (yrx). R is antisymmetric provided: x A, y A, (xry) (yrx) x = y. Can you visualize these properties in terms of the graph of R? Most relations are neither symmetric nor antisymmetric! Antisymmetric does not quite mean (xry) (y R x).

18 Transitive R is transitive provided: x A, y A, z A, (xry) (yrz) (xrz). You probably can t visualize this property in terms of the graph of R.

19 Examples Refl. Irref. Symm. A-S. Trans. = " % " " " < % " % " " " % % " " " % % % " % % " % %

20 A mystery property What would be a good name for the following property? R is a function provided: x A, y A, z A, (xry) (xrz) (y = z).