447
edits
Anisomorphism
Joined 24 January 2021
→Algebraic Geometry of Computing
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And statements written with logical connectives: <math> (x\and y)\or z = OR(AND(x,y),z) </math> | And statements written with logical connectives: <math> (x\and y)\or z = OR(AND(x,y),z) </math> | ||
<br> | |||
Along with distributive laws: <math> (x\and y)\or z = (x\or z)\and(y\or z) </math>, <math> (x\or y)\and z = (x\and z)\or(y\and z) </math> | |||
<br> | |||
De Morgan's laws: <math> \neg(x\and y) = (\neg x)\or (\neg y) </math>, <math> \neg(x\or y) = (\neg x)\and (\neg y) </math> | |||
<br> | |||
All of which apply to more complicated sentences rather than just individual variables. These laws along with commutative and associative laws are sufficient to evaluate and simplify any general logical expression, however we contend that this is the wrong language for computing and makes other important aspects - the dynamics and algebra - obscure. | |||
<br> | |||
There is one thing we can extract from logical connectives before moving on. The disjunctive normal form allows us to read truth tables and directly translate them into connective formulae which we can use later. Let us look at a different example which will help us escape the artificiality of AND and OR. | |||
{| class="wikitable" style="margin:auto" | |||
|+ XOR | |||
|- | |||
! x !! y !! XOR(x,y) | |||
|- | |||
| 0 || 0 || 0 | |||
|- | |||
| 0 || 1 || 1 | |||
|- | |||
| 1 || 0 || 1 | |||
|- | |||
| 1 || 1 || 0 | |||
|} | |||
XOR is only "true" or 1 when x or y but not both, are 1. | |||
= Read prototype = | = Read prototype = | ||