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Chapter 3 Review MATH130 Heidi Burgiel Relation • A relation R from X to Y is any subset of XxY • 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 3. 12345 110010 201001 300100 410010 501001 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 3.1.21 – 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. 11001 01001 00101 00011 00001 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. 11001 11001 12003 01001 01001 01002 00101x00101 = 00102 00011 00011 00012 00001 00001 00001 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 Si where xRy if x and y are in the same subset Si. 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. 11000 11000 00100 00011 00011