【Basic Abstract Algebra】Exercises for Section 1.4 — Factor Sets
1.【Basic Abstract Algebra】Exercises of Section 1.1 — Sets2.【Basic Abstract Algebra】Exercises for Section 1.2 — Maps3.【Basic Abstract Algebra】Exercises for Section 1.3 — Equivalence relation and equivalence classes
4.【Basic Abstract Algebra】Exercises for Section 1.4 — Factor Sets
5.【Basic Abstract Algebra】Exercises for Section 1.5 — Number Theory6.【Basic Abstract Algebra】Exercises for Section 1.6 — The Chinese Remainder Theorem7.【Basic Abstract Algebra】Exercises for Section 2.1 — Definitions and examples8.【Basic Abstract Algebra】Exercises for Section 2.2 — Subgroups9.【Basic Abstract Algebra】Exercises for Section 2.3 — Cyclic groups10.【Basic Abstract Algebra】Exercises for Section 2.4 — Permutation groups11.【Basic Abstract Algebra】Exercises for Section 2.5 — Dihedral groups12.【Basic Abstract Algebra】Exercises for Section 3.1 — Cosets and Lagrange's Theorem13.【Basic Abstract Algebra】Exercises for Section 3.2 — Normal subgroups and factor groups14.【Basic Abstract Algebra】Exercises for Section 3.3 — Homomorphism of groups15.【Basic Abstract Algebra】Exercises for Section 3.5 — Fundamental Isomorphism theorem of group- 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:
x∣x is true for any x∈X, so R is reflexive. - Symmetry:
If x∣y, it does not necessarily imply y∣x. For example, if x=2 and y=4, 2∣4, 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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