RELATIONS. Relation of Students to Courses:

Transcription

1 RELATIONS A relation can be thought of as a set of ordered pairs. We consider the first element of the ordered pair to be related to the second element of the ordered pair. Relation of Students to Courses: Student Bill Mary Bill Ron Ron Dave Course CompSci Math Art History CompSci Math Definition: A relation R from a set X to a set Y is a subset of the Cartesian Product X x Y, if (x, y) R, we write x R y and say that x is related to y. In case X = Y, we call R a binary relation on X.

2 Domain and Range of R The set {x X (x, y) for some y Y} is called the domain of R. The set {y Y (x, y) for some x X} is called the range of R. If a relation is given as a table, the domain consists of the first column and the range consists of the second column. Example: Our relation R in the table above can be re-written as this set of ordered pairs: R= { (Bill, CompSci), (Mary, Math), (Bill, Art), (Ron, History), (Ron, CompSci), (Dave, Math) } Since (Dave, Math) R, we may write that Dave R Math. The domain is X= {Bill, Mary, Ron, Dave}. The range is Y = {CompSci, Math, Art, History} Relation defined by rule of membership: Example 1: Let us define two sets X and Y such that:

3 X = {2, 3, 4}, Y ={3, 4, 5, 6, 7} If we define a relation R from X to Y by: (x, y) R if x divides y (with zero remainder). We obtain that the following ordered pairs belong to R: R = {(2,4), (2, 6), (3, 3), (3, 6), (4, 4)} If we write R as a table, we obtain: X Y The domain of R is the set {2, 3, 4} and the range of R is the set {3, 4, 6} Example 2: Let R be the relation on X = {1, 2, 3, 4} defined by (x, y) R if x y, for x, y X. then R = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 3), (2, 4), (3, 3), (3, 4), (4, 4)} The domain and range of R are both equal to X.

4 Digraph: An informative way to picture a relation on a set is to draw its digraph. To draw the digraph of a relation on a set X: 1. First, draw dots or vertices to represent the elements of X. 2. Next, if the ordered pair (x, y) R, draw an arrow, called a directed edge from x to y. 3. An element of the form (x, x) is in relation with itself and corresponds to a directed edge from x to x called a loop Example 3: The relation R on X = {a, b, c, d} is given by the digraph R= {(a, a), (b, c), (c, b), (d, d) } Properties of Relations:

5 Reflexive: A relation R on a set X is called reflexive if (x, x) R for every x X. The digraph of a reflexive relation has a loop on every vertex. The relation R on X = {1, 2, 3, 4}( in example 2), is reflexive because: for each element x X, (x, x) R. The relation R on X = { a, b, c, d} (in example 3) is not reflexive. For example, b X but (b, b) R. Symmetric: A relation R on a set X is called symmetric if : for all x, y X, if (x, y) R then (y, x) R. The digraph of a symmetric relation has the property that whenever there is a directed edge from any vertex v to a vertex w, then there is a directed edge from w to v. The relation R on X = { a, b, c, d} (in example 3) is symmetric because: for all x, y X, if (x, y) R then (y, x) R

6 The relation R on X = {1, 2,3,4} (in example 2) is not symmetric because: For example (2, 3) R but (3, 2) R. Antisymmetric: A relation R on a set X is called antisymmetric if for all x, y X, if (x, y) R and x y then (y, x) R. The digraph of an antisymmetric relation has the property that between any two vertices there is at most one directed edge. The relation R in example 2 is antisymmetric. The relation R in example 3 is not antisymmetric because both (b,c) and (c,b) are in R If a relation R on X has no members of the form (x, y) with x y, then R is antisymmetric. For example: If X = {1, 2, 3} and we define the relation R On X such that : (x, y) R if x = y R= { (1, 1), (2, 2), (3, 3)} This relation R on X is antisymmetric. R is also reflexive and symmetric.

7 Transitive: A relation R on a set X is called transitive if: for all x, y, z X, if (x, y) R and (y, z) R then (x, z) R The relation R in example 2 is transitive because for all x, y, z X, if (x, y) R and (y, z) R then (x, z) R To formally verify that that this relation satisfies the definition, we would have to list all the pairs of form (x, y) and (y, z) and verify that (x, z) R. We do not need to verify that the relation is true in case x=y or y=z. (x, y) (y, z) (x, z) (1, 2) (2, 3) (1, 3) (1, 2) (2, 4) (1, 4) (1, 3) (3, 4) (1, 4) (2, 3) (3, 4) (2, 4) The digraph of a transitive relation has the property that whenever there are directed edges from x to y and from y to z, there is also a directed edge from x to z. The relation in example 3 is not transitive, because (b, c) and (c, b) R but (b, b) R.

8 Partial Orders: A relation R on a set X is called a partial order if R is reflexive, antisymmetric and transitive. For example, the relation R defined on the set of integers by: (x, y) R if x y is a partial order, it orders the integers