Functions, Relations, Partial Order Relations Definition: The Cartesian product of two sets A and B (also called the product set, set direct product, or cross product) is defined to be the set of all pairs (a,b) where a ∈ A and b ∈ B . It is denoted A X B. Example : A ={1, 2), B= { a, b} A X B = { (1, a), (1, b), (2, a), (2, b)} Solve the following: X = {a, c} and Y = {a, b, e, f}. Write down the elements of: (a) X × Y (b) Y × X (c) X 2 (= X × X) (d) What could you say about two sets A and B if A × B = B × A? Definition: A function is a subset of the Cartesian product.A relation is any subset of a Cartesian product. For instance, a subset of , called a "binary relation from to ," is a collection of ordered pairs with first components from and second components from , and, in particular, a subset of is called a "relation on ." For a binary relation , one often writes to mean that is in . If (a, b) ∈ R × R , we write x R y and say that x is in relation with y. Exercise 2: A chess board’s 8 rows are labeled 1 to 8, and its 8 columns a to h. Each square of the board is described by the ordered pair (column letter, row number). (a) A knight is positioned at (d, 3). Write down its possible positions after a single move of the knight. Answer: (b) If R = {1, 2, ..., 8}, C = {a, b, ..., h}, and P = {coordinates of all squares on the chess board}, use set notation to express P in terms of R and C. Answer: (c) A rook is positioned at (g, 2). If T = {2} and G = {g}, express its possible positions after one move of the rook in terms of R, C, T and G. Look up: “domain of relation and range of a relation” and define the domain and range of the relation described in 2- Definition Binary function: If a relation is defined for X to X, then the relation R is called binary. In case X = Y, we call R a _______________ on X. Let R2 be the relation on X = {1, 2, 3, 4} defined by (x, y) ∈ R2 if x ≤ y, for x, y ∈X. then R2 = { (1,1),(1,2), (1,3),(1,4),(2,3),(2,4),(2,2),(3,3),(4,4),(3,4)} D(R2)= R(R2) = Diagraph: The diagraph of a relation is a graphical representation of the relation. 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. Draw the diagraph for the relation R2. The relation R3 on X = {a, b, c, d} defined as R= {(a, a), (b, c), (c, b), (d, d) }. Draw its digraph. 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. • 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. • 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. • 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 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. Exercise: Find if R2 is reflexive, symmetric, antisymmetric and transitive. Same question for R3. Let X = {1, 2, 3} and we define the relation R on X such that: (x, y) ∈ R if x = y. Find if this relation is reflexive, symmetric, antisymmetric and transitive. .