8: Maximum and Minimum

Datamatrix 2d Barcode barcode library with .netuse visual studio .net 2d data matrix barcode generator toattach data matrix barcode for .net

Second and third, we use the fact that true is a unit of conjunction and, again, ^ is reflexive by instantiating z to x and z to y we get, respectively,

2d Data Matrix Barcode barcode library in .netUsing Barcode recognizer for .net vs 2010 Control to read, scan read, scan image in .net vs 2010 applications.

x\y ^ x =

Generate barcode in .netuse vs .net barcode integrating toencode barcode for .net

x\y ^ y =

recognize barcode on .netUsing Barcode recognizer for visual .net Control to read, scan read, scan image in visual .net applications.

A fourth property can be obtained from the fact that ^ is a total ordering. That is, for all x and y, x^y v y^x . Using this fact, we calculate: x\y = x v x\y = y { = antisymmetry of ^ } (x\y ^ x A x^x\y) v (x\y ^ y A y^xly) { above, substitution of equals for equals } A true) v (x^y A true) { true is the unit of conjunction } v x^y = { true . So maximum is a choice function: x\y chooses between x and y. Here are a couple of very short exercises. Exercise 8.2. Show that x\y ^ x = x\y = x. Exercise 8.3. Simplify the statement x\y ^ x+y. D D ^ is a total ordering }

8.2 Using Indirect Equality

Control data matrix 2d barcode size on .netto receive data matrix ecc200 and gs1 datamatrix barcode data, size, image with .net barcode sdk

To derive additional equalities, we recall the rule of indirect equality: in order to establish the equality x = y , show that, for arbitrary z of the same type as x and

Control datamatrix 2d barcode image in visual basicusing visual studio .net todeploy data matrix barcode for asp.net web,windows application

Here are two examples. In the first example, we show that maximum is associative because conjunction is associative. We have, for all w, definition of max }

Visual Studio .NET Crystal matrix barcode drawer on .netusing visual .net crystal topaint matrix barcode in asp.net web,windows application

8.2 Using Indirect Equality

Barcode Pdf417 barcode library for .netuse visual .net crystal pdf417 2d barcode integrated todraw pdf417 2d barcode on .net

{ { {

Bar Code barcode library on .netusing .net vs 2010 crystal toget barcode in asp.net web,windows application

definition of max

Linear 1d Barcode barcode library for .netusing visual studio .net crystal togenerate 1d for asp.net web,windows application

A y^w) A z^w

Use identcode with .netusing vs .net toattach identcode on asp.net web,windows application

A is associative } A (y^w A z^w) definition of max (applied twice) }

UPCA writer for visual c#.netusing asp.net web pages crystal toinclude upc-a supplement 5 on asp.net web,windows application

Thus, by indirect equality, (x\y}\z = x\(y\z] . Note how short and straightforward this proof is. In contrast, if maximum is defined in the conventional way by case analysis, it would be necessary to consider six different cases, six being the number of ways to order three values x, y and z. In the second example, we derive a distributivity property of maximum. We have, for all w,

Asp.net Webform Crystal qr code 2d barcode printing for vb.netusing barcode integration for aspx crystal control to generate, create qr-codes image in aspx crystal applications.

x + (y\z] ^ w

Control barcode pdf417 size for microsoft wordto paint pdf-417 2d barcode and pdf 417 data, size, image with word barcode sdk

{ y\z ^ w-x = = { definition of max } y ^w-x A z ^w-x { shunt 'x +' back in order to be able to apply the definition of maximum } x + y^w A x + z^w { definition of max } (x+y}\(x+z) ^ w . Thus, by indirect equality, x + (y\z] = (x+y)\(x+z] . Try the following for yourself, observing carefully the properties of conjunction that you exploit. Exercise 8.4. Prove the following. (a) x\x = x . (b) x\y = y\x . D = shunt 'x +' out of the way in order to be able to apply the definition of maximum }

Matrix Barcode integrated on .netusing windows forms tobuild 2d matrix barcode on asp.net web,windows application

8: Maximum and Minimum Another property we can derive from the definition is by using contraposition: true = = { { { = { definition of maximum } contrapositive } De Morgan } -i(w^v) = v <u } x t y s ^ z = x s=5 z A y s^ z ->(xty ^ z) = - - ( x ^ Z A y ->(xTy ^ z) = ->(x<z) v z<x\y = z<xvz<y . Thus we have derived

DataMatrix barcode library on visual basic.netgenerate, create barcode data matrix none for visual basic.net projects

z<x\y = z<xvz<y .

SQL 2008 code128 implement in .netusing sql server reporting service toadd code-128 for asp.net web,windows application

This too is a distributMty property: the function (z <) distributes over maximum turning it into a disjunction. Now we verify that the function (z ^) also distributes over maximum using the properties of disjunction as opposed to the properties of conjunction. v { { definition } rearrangement of terms (allowed because disjunction is symmetric and associative) }

Pdf417 barcode library with visual c#.netuse visual studio .net (winforms) crystal barcode pdf417 printer toconnect pdf417 with visual c#

(z = x v z = .y) v (z<x v z<y) (z = x v z<x) v (z = y v z<y)

Control qr codes size on .net qr code 2d barcode size on .net

in the second disjunct: z<x\y = z <x v z <y } See the calculation below. }

z = x v z = y v z<x\y

The last step in the above needs justification. On the face of it, it looks like the step is an easy one. After all, = z = x\y v z<x\y , so that it looks tike we just need to replace z = x v z = y b y z = xty. But it is not the case that z = x v z = y = z = x!yin general. A bit more thought is necessary.