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Understanding Relations in Mathematics: Definitions and Examples

Explore the concepts of relations in mathematics, including symmetric, antisymmetric, transitive, and equivalence relations, with detailed examples and matrices. Learn about partial orders and equivalence classes in this comprehensive guide.

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Understanding Relations in Mathematics: Definitions and Examples

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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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