Chapter 3 Review MATH 130 Heidi Burgiel Relation

Chapter 3 Review MATH 130 Heidi Burgiel

Relation • A relation R from X to Y is any subset of Xx. Y • The matrix of a Relation R is a matrix that has a 1 in row x and column y whenever x. Ry (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 x. Ry 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 x. Rx for all elements x of X.

Examples of Antisymmetric Relations • x. Ry if x < y • x. Ry if x is a subset of y • x. Ry 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 x. Ry and y. Rz, it is also true that x. Rz. • Examples: x. Ry if x=y x. Ry if x<y x. Ry if step x must occur before step y

Partial Order • A relation that is reflexive, antisymmetric and transitive is a partial order. • Examples: x. Ry if x<y x. Ry 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 00101 00011 00001

The matrix of a transitive relation • If M is the matrix of a transitive relation, then the matrix Mx. M has no more zeros than matrix M. 11001 12003 01001 01002 00101 x 00101 = 00102 00011 00012 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 x. Ry 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 00100 00011