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 3.

1 2 3 4 5 1 1 0 0 1 0 2 0 1 0 0 1 3 0 0 1 0 0 4 1 0 0 1 0 5 0 1 0 0 1

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

Partial Order

• A relation that is reflexive, antisymmetric and transitive is a partial order.

• Examples: xRy if x

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.

1 1 0 0 1 0 1 0 0 1 0 0 1 0 1 0 0 0 1 1 0 0 0 0 1

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.

1 1 0 0 1 1 1 0 0 1 1 2 0 0 3 0 1 0 0 1 0 1 0 0 1 0 1 0 0 2 0 0 1 0 1 x 0 0 1 0 1 = 0 0 1 0 2 0 0 0 1 1 0 0 0 1 1 0 0 0 1 2 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1

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.

1 1 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 1 1