Application( Cont. Example(Equivalence relation) The equivalence relation can be formalized with a single binary relation symbols as follows: Solution (XR(x, x), (x)(y(R(x,y)>R(, x) (x)(y)(Vz)(R(x,y)∧R(y,z))→R(x,z)Application(Cont.) Example (Equivalence relation) The equivalence relation can be formalized with a single binary relation symbols as follows: Solution. Φequ =    (∀x)R(x, x), (∀x)(∀y)(R(x, y) → R(y, x), (∀x)(∀y)(∀z)((R(x, y) ∧ R(y, z)) → R(x, z)). Yi Li (Fudan University) Discrete Mathematics June 20, 2012 8 / 15