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vb.net code 128 font Discrete Mathematics Demystified in Java
Discrete Mathematics Demystified Create UPCA Supplement 5 In Java Using Barcode generator for Java Control to generate, create Universal Product Code version A image in Java applications. GS1  12 Scanner In Java Using Barcode scanner for Java Control to read, scan read, scan image in Java applications. EXAMPLE 54 The equivalence class [(4, 12)] contains all of the pairs (4, 12), (1, 3), ( 2, 6) (Of course it contains in nitely many other pairs as well) This equivalence class represents the fraction 4/12 which we sometimes also write as 1/3 or ( 2)/( 6) If [(a, b)] and [(c, d)] are rational numbers then we de ne their product to be the rational number [(a c, b d)] This is well de ned (unambiguous), for the following reason Suppose that (a, b) is related to (a, b) and (c, d) is related to (c, d) We would like to know that [(a, b)] [(c, d)] = [(a c, b b)] is the same equivalence class as [(a, b)] [(c, d)] = [(a c, b d)] In other words we need to know that (a c) (b d) = (a c) (b d) But our hypothesis is that a b =a b and c d =c d (52) Bar Code Maker In Java Using Barcode maker for Java Control to generate, create barcode image in Java applications. Barcode Decoder In Java Using Barcode reader for Java Control to read, scan read, scan image in Java applications. Multiplying together the left sides and the right sides we obtain (a b) (c d) = (a b) (c d) Rearranging, we have (a c) (b d) = (a c) (b d) But this is just Eq 52 So multiplication is unambiguous EXAMPLE 55 The product of the two rational numbers [(3, 8)] and [( 2, 5)] is [(3 ( 2), 8 5)] = [( 6, 40)] = [( 3, 20)] This is what we expect: the product of 3/8 and 2/5 is 3/20 If q = [(a, b)] and r = [(c, d)] are rational numbers and if r is not zero (that is, [(c, d)] is not the equivalence class zero in other words, c = 0) then we de ne Making UPCA Supplement 2 In Visual C# Using Barcode drawer for .NET Control to generate, create UPC Symbol image in Visual Studio .NET applications. UPCA Drawer In .NET Framework Using Barcode printer for ASP.NET Control to generate, create UCC  12 image in ASP.NET applications. Number Systems
Painting GS1  12 In VS .NET Using Barcode creator for .NET Control to generate, create GS1  12 image in .NET applications. Print UPCA In Visual Basic .NET Using Barcode creator for .NET framework Control to generate, create Universal Product Code version A image in .NET applications. the quotient q/r to be the equivalence class [(ad, bc)] We leave it to you to check that this operation is well de ned EXAMPLE 56 The quotient of the rational number [(4, 7)] by the rational number [(3, 2)] is, by de nition, the rational number [(4 ( 2), 7 3)] = [( 8, 21)] This is what we expect: the quotient of 4/7 by 3/2 is 8/21 How should we add two rational numbers We could try declaring [(a, b)] + [(c, d)] to be [(a + c, b + d)], but this will not work (think about the way that we usually add fractions) Instead we de ne [(a, b)] + [(c, d)] = [(a d + b c, b d)] That this de nition is well de ned (unambiguous) is left for the exercises We turn instead to an example EXAMPLE 57 The sum of the rational numbers [(3, 14)] and [(9, 4)] is given by [(3 4 + ( 14) 9, ( 14) 4)] = [( 114, 56)] = [(57, 28)] This coincides with the usual way that we add fractions: 9 57 3 + = 14 4 28 Draw Data Matrix 2d Barcode In Java Using Barcode generator for Java Control to generate, create Data Matrix image in Java applications. Generate Code 128C In Java Using Barcode maker for Java Control to generate, create Code 128A image in Java applications. Notice that the equivalence class [(0, 1)] is the rational number that we usually denote by 0 It is the additive identity, for if [(a, b)] is another rational number then [(0, 1)] + [(a, b)] = [(0 b + 1 a, 1 b)] = [(a, b)] A similar argument shows that [(0, 1)] times any rational number gives [(0, 1)] or 0 By the same token the rational number [(1, 1)] is the multiplicative identity We leave the details for you Of course the concept of subtraction is really just a special case of addition [that is, a b is the same as a + ( b)] So we shall say nothing further about subtraction Create EAN / UCC  13 In Java Using Barcode encoder for Java Control to generate, create UCC128 image in Java applications. EAN13 Drawer In Java Using Barcode generator for Java Control to generate, create GS1  13 image in Java applications. Discrete Mathematics Demystified
Printing Uniform Symbology Specification ITF In Java Using Barcode creator for Java Control to generate, create Interleaved 2 of 5 image in Java applications. Barcode Creator In .NET Using Barcode maker for ASP.NET Control to generate, create bar code image in ASP.NET applications. In practice we will write rational numbers in the traditional fashion: 2 19 22 24 , , , , 5 3 2 4 In mathematics it is generally not wise to write rational numbers in mixed form, such as 2 3 , because the juxtaposition of two numbers could easily be mistaken for 5 multiplication Instead we would write this quantity as the improper fraction 13/5 De nition 51 A set S is called a eld if it is equipped with a binary operation (usually called addition and denoted + ) and a second binary operation (usually called multiplication and denoted ) such that the following axioms are satis ed: A1 S is closed under addition: if x, y S then x + y S A2 Addition is commutative: if x, y S then x + y = y + x A3 Addition is associative: if x, y, z S then x + (y + z) = (x + y) + z A4 There exists an element, called 0, in S which is an additive identity: if x S then 0 + x = x A5 Each element of S has an additive inverse: if x S then there is an element x S such that x + ( x) = 0 M1 S is closed under multiplication: if x, y S then x y S M2 Multiplication is commutative: if x, y S then x y = y x M3 Multiplication is associative: if x, y, z S then x (y z) = (x y) z M4 There exists an element, called 1, which is a multiplicative identity: if x S then 1 x = x M5 Each nonzero element of S has a multiplicative inverse: if 0 = x S then there is an element x 1 S such that x (x 1 ) = 1 The element x 1 is sometimes denoted by 1/x D1 Multiplication distributes over addition: if x, y, z S then x (y + z) = x y + x z Eleven axioms is a lot to digest all at once, but in fact these are all familiar properties of addition and multiplication of rational numbers that we use every day: the set Q, with the usual notions of addition and multiplication (and with the usual additive identity 0 and multiplicative identity 1), forms a eld The integers, by contrast, do not: nonzero elements of Z (except 1 and 1) do not have multiplicative inverses in the integers Let us now consider some consequences of the eld axioms Generating Data Matrix In None Using Barcode printer for Microsoft Word Control to generate, create Data Matrix image in Microsoft Word applications. Generating Barcode In Java Using Barcode generator for BIRT Control to generate, create bar code image in BIRT applications. Paint EAN / UCC  14 In None Using Barcode maker for Online Control to generate, create EAN128 image in Online applications. Linear Barcode Generator In .NET Using Barcode generator for ASP.NET Control to generate, create 1D image in ASP.NET applications. Making Barcode In Java Using Barcode maker for Android Control to generate, create barcode image in Android applications. Generating Code 128A In VB.NET Using Barcode creation for Visual Studio .NET Control to generate, create Code128 image in VS .NET applications. 
