# 等价关系

（重定向自等價關係

1. 自反性：${\displaystyle \forall x\in A,~~xRx}$
2. 对称性：${\displaystyle \forall x,y\in A,~~xRy~~\implies ~~yRx}$
3. 传递性：${\displaystyle \forall x,y,z\in A,~~~(xRy~~\wedge ~~yRz)~~\implies ~~xRz}$

${\displaystyle xRy\iff \forall x,y\in A,~x\equiv y{\pmod {3}}}$

• 沒有自反性：任何一個數不能比自身為較大（${\displaystyle n\ngtr n}$
• 沒有對稱性：如果${\displaystyle m>n}$，就肯定不能有${\displaystyle n>m}$

### 不是等价关系的关系的例子

• 实数之间的"≥"关系满足自反性和传递性，但不满足对称性。例如，7 ≥ 5 无法推出 5 ≥ 7。它是一种全序关系

## 參考文獻

• Brown, Ronald, 2006. Topology and Groupoids. Booksurge LLC. ISBN 1-4196-2722-8.
• Castellani, E., 2003, "Symmetry and equivalence" in Brading, Katherine, and E. Castellani, eds., Symmetries in Physics: Philosophical Reflections. Cambridge Univ. Press: 422-433.
• and Crawley, Peter, 1973. Algebraic Theory of Lattices. Prentice Hall. Chpt. 12 discusses how equivalence relations arise in lattice theory.
• Higgins, P.J., 1971. Categories and groupoids.页面存档备份，存于互联网档案馆 Van Nostrand. Downloadable since 2005 as a TAC Reprint.
• , 1973. A Treatise on Time and Space. London: Methuen. Section 31.
• Rosen, Joseph (2008) Symmetry Rules: How Science and Nature are Founded on Symmetry. Springer-Verlag. Mostly chpts. 9,10.
• (1965) Introduction to the Foundations of Mathematics 2nd edition, Chapter 2-8: Axioms defining equivalence, pp 48–50, John Wiley & Sons.