等价关系(equivalence relation)即设是某个集合上的一个二元关系。若满足以下条件:
- 自反性:
- 对称性:
- 传递性:
则称是一个定义在上的等价关系。习惯上会把等价关系的符号由改写为。
例如,设,定义上的关系如下:
其中叫做与模3 同余,即除以3的余数与除以3的余数相等。例子有1R4, 2R5, 3R6。不难验证为上的等价关系。
并非所有的二元关系都是等价关系。一个简单的反例是比较两个数中哪个较大:
- 没有自反性:任何一个数不能比自身为较大()
- 没有对称性:如果,就肯定不能有
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。
参见
参考文献
- 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.
外部链接
- 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.