等价关系
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等價關係（equivalence relation）即设Rdisplaystyle R
- 自反性：∀x∈A, xRxdisplaystyle forall xin A,~~xRx
- 对称性：∀x,y∈A, xRy ⟹ yRxdisplaystyle forall x,yin A,~~xRy~~implies ~~yRx
- 传递性：∀x,y,z∈A, (xRy ∧ yRz) ⟹ xRzdisplaystyle forall x,y,zin A,~~~(xRy~~wedge ~~yRz)~~implies ~~xRz
则称Rdisplaystyle R
例如，设A=1,2,…,8displaystyle A=1,2,ldots ,8
- xRy⟺∀x,y∈A, x≡y(mod3)displaystyle xRyiff forall x,yin A,~xequiv ypmod 3
其中x≡y(mod3)displaystyle xequiv ypmod 3
并非所有的二元關係都是等價關係。一個簡單的反例是比較兩個數中哪個較大：
- 沒有自反性：任何一個數不能比自身為較大（n≯ndisplaystyle nngtr n
）
- 沒有對稱性：如果m>ndisplaystyle m>n
，就肯定不能有n>mdisplaystyle n>m
参见
- 等价类
- 集合划分
- 商集
- 等价符号
- Apartness relation
- 共轭类
- Equipollence (geometry)
- Topological conjugacy
- Up to
參考文獻
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- 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.
Robert Dilworth 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. Van Nostrand. Downloadable since 2005 as a TAC Reprint.
John Randolph Lucas, 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.
Raymond Wilder (1965) Introduction to the Foundations of Mathematics 2nd edition, Chapter 2-8: Axioms defining equivalence, pp 48–50, John Wiley & Sons.
外部連結
Hazewinkel, Michiel (编), Equivalence relation, 数学百科全书, Springer, 2001, ISBN 978-1-55608-010-4- Bogomolny, A., "Equivalence Relationship" cut-the-knot. Accessed 1 September 2009
Equivalence relation at PlanetMath
Binary matrices representing equivalence relations at OEIS.
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