Chapter 3 Review - Webhost Bridgew

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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

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