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【Basic Abstract Algebra】Exercises for Section 1.4 — Factor Sets

  1. Let X be a set and R is a relation X. Define xRy if x|y. Is R an equivalence relation, partial ordering relation or totally ordering relation?

Solution:
(1) equivalence?

  • Reflexivity:
    xx is true for any xX, so R is reflexive.
  • Symmetry:
    If xy, it does not necessarily imply yx. For example, if x=2 and y=4, 24, but 4. Thus, R is not symmetric.
  • Transitivity:
    If x\mid y,~y\mid z\Rightarrow x\mid z, so R is transitive.

Since R is not symmetric, it is not an equivalence relation.

(2) partial ordering relation?

  • Reflexivity:

    As shown earlier, R is reflexive.

  • Antisymmetry:
    If x \mid y and y \mid x, then x = y (since x \mid y and y \mid x imply x and y have the same absolute value). Thus, R is antisymmetric.

  • Transitivity:
    As shown earlier, R is transitive.

    Since R satisfies reflexivity, antisymmetry, and transitivity, it is a partial ordering relation.

(3) totally ordering relation?

  • For some x, y \in X, neither x \mid y nor y \mid x may hold. For example, if x = 2 and y = 3, 2 \nmid 3 and 3 \nmid 2. Thus, R is not a total ordering relation.
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