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Chapter 3 Review

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.

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Chapter 3 Review

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  1. Chapter 3 Review MATH130 Heidi Burgiel

  2. 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.

  3. 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

  4. 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.

  5. 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

  6. 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

  7. 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

  8. 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

  9. 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

  10. 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.

  11. 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

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