Definition 34: We may use the following recursively defined notation when working with binary predicates. Recall the usual notation for conjunction.
(a) Base cases: For binary predicates \sim_A and \sim_B , we use the abbreviation (a \sim_A b \sim_B c) \hspace{3mm} \textup{for} \hspace{3mm} \big( (a \sim b) \wedge (b \sim_B c)\big). (b) Recursion: Given binary predicates \sim_A, \sim_B, ..., \sim_X, \sim_Y such that (a \sim_A b \sim_B \cdots \sim_X y) is understood, we use the abbreviation (a \sim_A b \sim_B \cdots \sim_X y \sim_Y z) \hspace{3mm} \textup{for} \hspace{3mm} \big( (a \sim_A \sim_B \cdots \sim_X y) \wedge (y \sim_Y z) \big).
On the other hand, given a binary predicate \sim , we may use the following recursively defined notation:
(a) Base cases: We understand (a, b \sim z) to mean \big( (a \sim z) \wedge (b \sim z) \big) .
(b) Recursion: If (a, b, ..., x \sim z) is understood then we further write (a, b, ..., x, y \sim z) \hspace{3mm} \textup{for} \hspace{3mm} \big( (a, b, ..., x \sim z) \wedge (y \sim z) \big).