## Friday, September 23, 2011

### EQ 1: Precedences

Precedences (from high to low)

(1) [x := e] (Textual Substitution)

(2) . (function application)

(3) Unary Prefix Operators + - $$\neg$$ # ~ $$\mathcal { P }$$

(4) **

(5) $$\ast$$ / $$\div$$ mod gcd

(6) + - $$\cup$$ $$\cap$$ $$\times$$ $$\circ$$ $$\bullet$$

(7) $$\downarrow$$ $$\uparrow$$

(8) #

(9) $$\triangleright$$ $$\triangleleft$$ ^

(10) = < > $$\epsilon$$ $$\subset$$ $$\subseteq$$ $$\supset$$ $$\supseteq$$ |

(11) $$\vee$$ $$\wedge$$

(12) $$\Rightarrow$$ $$\Leftarrow$$

(13) $$\equiv$$

Notes: All non associative prefix binary operators associate to the left, except ** , $$\triangleleft$$ and $$\Rightarrow$$ which associate to the right.

All operators on lines 10, 12 and 13 may have a slash / through them to denote negation.

Thus b $$\not\equiv$$ c is equivalent to $$\neg$$ ( b $$\equiv$$ c )

A core subset of the above, omitting unnecessary symbols, which is used to prove the theorems of the logic itself is as follows

(a) [x := e] (Textual Substitution)

(b) . (function application)

(c) $$\neg$$

(d) $$\ast$$ /

(e) =

(f) $$\vee$$ $$\wedge$$

(g) $$\Rightarrow$$ $$\Leftarrow$$

(h) $$\equiv$$

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