Chapter 3 Review MATH130 Heidi Burgiel

Relation A relation R from X to Y is any subset of X x Y The matrix of a Relation R is a matrix that has a 1 in row x and column y whenever xRy (if (x, y) is in R) and otherwise has a 0 in row x, column y.

Example X = {1, 2, 3, 4, 5}, R is a binary relation on X defined by xRy if x mod 3 = y mod

Symmetric, Reflexive, Antisymmetric A relation R on X is symmetric if its matrix is symmetric – in other words, if whenever (x,y) is in R, (y,x) is in R. A relation R on X is antisymmetric if whenever (x,y) is in R and x ≠ y, (y,x) is not in R. A relation R on X is reflexive if xRx for all elements x of X.

Examples of Antisymmetric Relations xRy if x < y xRy if x is a subset of y xRy if step x has to happen before step y In the matrix of an antisymmeric relation, if there is a 1 in position i,j then there is a 0 in position j,i

Transitive A relation is transitive if whenever xRy and yRz, it is also true that xRz. Examples: xRy if x=y xRy if x<y xRy if step x must occur before step y

Partial Order A relation that is reflexive, antisymmetric and transitive is a partial order. Examples: xRy if x<y xRy if step x must occur before step y

Matrix of a Partial Order Example – using a camera When the elements of X are put in order, the matrix of a relation that is a partial order looks upper triangular

The matrix of a transitive relation If M is the matrix of a transitive relation, then the matrix MxM has no more zeros than matrix M x =

Equivalence Relation A relation R on X is an equivalence relation if it is symmetric, transitive and reflexive. An equivalence relation groups the elements of X into disjoint subsets S i where xRy if x and y are in the same subset S i. The set of all these subsets is a partition of X.

Matrix of an equivalence relation If the elements of X are ordered correctly, the matrix of an equivalence relation looks like a collection of squares of 1’s