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 Course Bill CompSci Mary Math Bill Art Ron History Ron CompSci Dave 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. 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: 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 2 4 2 6 3 3 3 6 4 4 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. 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. 1. 2 . 3 4 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: 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 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. 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. 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 .