EQ 4 Equality vs Equivalence

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)