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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. - PowerPoint PPT Presentation
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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 51 1 0 0 1 02 0 1 0 0 13 0 0 1 0 04 1 0 0 1 05 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<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.
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 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.
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