b \( \equiv\) c is evaluated the same as b = c (in terms of truth tables) except that \( \equiv \) is allowed only when b and c are booleans.

Definition of Conjunctional

If \( \circ \) and \( \star \) are conjunctional operators, b \( \circ \) c \( \star \) d is equivalent to b \( \circ \) c \( \wedge \) c \( \star \) d.

From the precedence table, all operators on line 10 are conjunctional, ie = < > \( \epsilon \) \( \subset \) \( \subseteq \) \( \supset \) \( \supseteq \) | are all conjunctional

Definition of Associative

Binary Operator \( \circ \) is associative \( \equiv \) ((b \( \circ \) c) \( \circ \) d) = (b \( \circ \) ( c \( \circ \) d ))

Key 1: \( \equiv \) is associative, = is conjunctional

Key 2: Conjunctional use of = (and other conjunctional operators ) are syntactic sugar.

Conversion back and forth:

\( \hspace{2 cm }\) b \( \equiv \) c \( \equiv \) d

= \( \hspace{0.5 cm } \) parenthesize

\( \hspace{2 cm }\) (b \( \equiv \) c ) \( \equiv \) d

= \( \hspace{0.5 cm } \) replace operator

\( \hspace{2 cm } \) (b = c) = d

and

\( \hspace{2 cm }\) b = c = d

= \( \hspace{0.5 cm } \) remove conjunctional syntactic sugar

\( \hspace{2 cm }\) b = c \( \wedge \) c = d

= \( \hspace{0.5 cm } \) parenthesize

\( \hspace{2 cm }\) (b = c) \( \wedge \) (c = d)

= \( \hspace{0.5 cm } \) replace operator

\( \hspace{2 cm }\) (b \( \equiv \) c) \( \wedge \) (c \( \equiv \) d)

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