predicates             package:relations             R Documentation

_R_e_l_a_t_i_o_n _P_r_e_d_i_c_a_t_e_s

_D_e_s_c_r_i_p_t_i_o_n:

     Predicate functions for testing for binary relations and
     endorelations, and special kinds thereof.

_U_s_a_g_e:

     relation_is_antisymmetric(x)
     relation_is_asymmetric(x)
     relation_is_bijective(x)
     relation_is_binary(x)
     relation_is_complete(x)
     relation_is_coreflexive(x)
     relation_is_endorelation(x)
     relation_is_equivalence(x)
     relation_is_functional(x)
     relation_is_injective(x)
     relation_is_irreflexive(x)
     relation_is_left_total(x)
     relation_is_linear_order(x)
     relation_is_partial_order(x)
     relation_is_reflexive(x)
     relation_is_right_total(x)
     relation_is_strict_linear_order(x)
     relation_is_strict_partial_order(x)
     relation_is_surjective(x)
     relation_is_symmetric(x)
     relation_is_tournament(x)
     relation_is_transitive(x)
     relation_is_weak_order(x)
     relation_is_preference(x)
     relation_is_preorder(x)
     relation_is_quasiorder(x)

_A_r_g_u_m_e_n_t_s:

       x: an object inheriting from class 'relation'.

_D_e_t_a_i_l_s:

     A binary relation is a relation with arity 2. An _endorelation_ R
     on a set X is a relation with domain D(R) = (X, X), i.e., a binary
     relation on X.

     Let us write x R y iff (x, y) is contained in R.

     A binary relation R is called

     _l_e_f_t-_t_o_t_a_l: for all x there is at least one y such that x R y.

     _r_i_g_h_t-_t_o_t_a_l: for all y there is at least one x such that x R y.

     _f_u_n_c_t_i_o_n_a_l: for all x there is at most one y such that x R y.

     _s_u_r_j_e_c_t_i_v_e: the same as right-total.

     _i_n_j_e_c_t_i_v_e: for all y there is at most one x such that x R y.

     _b_i_j_e_c_t_i_v_e: left-total, right-total, functional and injective.

     An endorelation R is called

     _r_e_f_l_e_x_i_v_e: x R x for all x.

     _i_r_r_e_f_l_e_x_i_v_e: there is no x such that x R x.

     _c_o_r_e_f_l_e_x_i_v_e: x R y implies x = y.

     _s_y_m_m_e_t_r_i_c: x R y implies y R x.

     _a_s_y_m_m_e_t_r_i_c: x R y implies that not y R x.

     _a_n_t_i_s_y_m_m_e_t_r_i_c: x R y and y R x imply that x = y.

     _t_r_a_n_s_i_t_i_v_e: x R y and y R z imply that x R z.

     _c_o_m_p_l_e_t_e: for all x and y, x R y or y R x.

     Some combinations of these basic properties have special names
     because of their widespread use:

     _p_r_e_o_r_d_e_r: reflexive and transitive.

     _q_u_a_s_i_o_r_d_e_r: the same as preorder.

     _e_q_u_i_v_a_l_e_n_c_e: a symmetric preorder.

     _w_e_a_k _o_r_d_e_r: complete and transitive.

     _p_r_e_f_e_r_e_n_c_e: the same as weak order.

     _p_a_r_t_i_a_l _o_r_d_e_r: an antisymmetric preorder.

     _s_t_r_i_c_t _p_a_r_t_i_a_l _o_r_d_e_r: irreflexive, transitive and antisymmetric.

     _l_i_n_e_a_r _o_r_d_e_r: a complete partial order.

     _s_t_r_i_c_t _l_i_n_e_a_r _o_r_d_e_r: a complete strict partial order.

     _t_o_u_r_n_a_m_e_n_t: complete and antisymmetric.

     If R is a weak order ("weak preference relation"), I = I(R)
     defined by x I y iff x R y and y R x is an equivalence, the
     _indifference relation_ corresponding to R.

     There seem to be no commonly agreed definitions for order
     relations: e.g., Fishburn (1972) requires these to be irreflexive.

_R_e_f_e_r_e_n_c_e_s:

     P. C. Fishburn (1972), _Mathematics of decision theory_. Methods
     and Models in the Social Sciences 3. Mouton: The Hague.

     H. R. Varian (2002), _Intermediate Microeconomics: A Modern
     Approach_. 6th Edition. W. W. Norton & Company.

