## Saturday, October 1, 2011

### 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)