 Home
 Products
 Integration
 Tutorial
 Barcode FAQ
 Purchase
 Company
Discrete Mathematics Demystified in Java
Discrete Mathematics Demystified Make UCC  12 In Java Using Barcode creation for Java Control to generate, create UPCA Supplement 2 image in Java applications. Decode GTIN  12 In Java Using Barcode decoder for Java Control to read, scan read, scan image in Java applications. postulate in Axiom 1 is a different property: it says that when we are combining three elements we may group them, two by two, in either of the two obvious ways; the same answer results A group that is commutative is called abelian in honor of Niels Henrik Abel (1802 1829) EXAMPLE 93 Let G be the positive real numbers and let the group operation be multiplication: P(x, y) = x y, where is ordinary multiplication of reals Then (G, P) is a group Axiom 1: Of course multiplication of real numbers is associative Axiom 2: The number 1 is the identity element for multiplication: 1 x = x 1 = x for any real number x Axiom 3: The multiplicative inverse of a group element is its ordinary reciprocal That is, if x R satis es x > 0 then 1/x is its multiplicative inverse EXAMPLE 94 Let G be the integers and let P(x, y) = x + y (ordinary addition) Then (G, P) is a group Axiom 1: Certainly addition of integers is associative Axiom 2: The number 0 is the additive identity Axiom 3: The additive inverse of a group element is its negative: if m Z then m is its group inverse EXAMPLE 95 Let G be the k k matrices with real entries and nonzero determinant This is sometimes called the general linear group on k letters and is denoted by GL(k, R) Let P be ordinary matrix multiplication Then (G, P) is a group Axiom 1: Matrix multiplication is associative Axiom 2: The group identity is the matrix 1 0 0 0 0 1 0 0 k Ik 0 0 1 0 0 0 0 1 k Thus, if m G, then Ik m = m Ik = m Create Bar Code In Java Using Barcode drawer for Java Control to generate, create bar code image in Java applications. Scan Bar Code In Java Using Barcode reader for Java Control to read, scan read, scan image in Java applications. Number Theory
UPCA Creator In Visual C#.NET Using Barcode generation for .NET Control to generate, create UPC Code image in Visual Studio .NET applications. GTIN  12 Printer In .NET Framework Using Barcode generation for ASP.NET Control to generate, create UPCA image in ASP.NET applications. Axiom 3: The multiplicative inverse of a group element is its matrix inverse Thus if m G then the inverse matrix m 1 is the group inverse Notice in this example that it is important to restrict attention to square matrices, so that multiplication of any two elements in any order will make sense We also require that each matrix have nonzero determinant, so that each matrix will have an inverse To see that G is closed under the group operation of matrix multiplication, we must note that if M, N G then det(M N ) = (det M)(det N ) = 0 Unlike the previous two examples, this last one is a noncommutative group The advantage of the axiomatic method, in the present context, is that when we prove a proposition or theorem about a group G, it applies simultaneously to all groups Thus the axiomatic method gives us both a way of being concise and a way of cutting to the heart of the matter Proposition 91 The multiplicative identity for a group is unique Proof: Then Let G be a group Let e and e both be elements of G that satisfy Axiom 2 e =e e =e Thus e and e must be the same group element Proposition 92 Let G be a group and g G Then there is only one multiplicative inverse for g Proof: Suppose that h and k both satisfy the properties of the multiplicative inverse (Axiom 3) relative to g Then h = h e = h (g k) = (h g) k = e k = k Thus h and k must be the same group element, establishing that the multiplicative inverse is unique Proposition 93 Let g be an element of the group G Then (g 1 ) 1 = g Proof: Observe that g g 1 = e and g 1 g = e Making UPCA Supplement 2 In .NET Framework Using Barcode maker for .NET Control to generate, create UPCA Supplement 2 image in VS .NET applications. Creating GS1  12 In VB.NET Using Barcode creation for .NET Control to generate, create UPCA Supplement 5 image in VS .NET applications. Data Matrix Creator In Java Using Barcode generator for Java Control to generate, create Data Matrix 2d barcode image in Java applications. UCC  12 Encoder In Java Using Barcode creation for Java Control to generate, create UCC  12 image in Java applications. UCC.EAN  128 Drawer In Java Using Barcode creator for Java Control to generate, create GTIN  128 image in Java applications. GS1128 Generator In Java Using Barcode generator for Java Control to generate, create GS1 128 image in Java applications. GTIN  14 Creation In Java Using Barcode creator for Java Control to generate, create UCC  14 image in Java applications. Code39 Creation In VS .NET Using Barcode drawer for .NET framework Control to generate, create Code 39 image in Visual Studio .NET applications. Paint Barcode In ObjectiveC Using Barcode generator for iPhone Control to generate, create barcode image in iPhone applications. Code 128 Code Set B Generation In C# Using Barcode generation for VS .NET Control to generate, create Code128 image in VS .NET applications. Universal Product Code Version A Maker In None Using Barcode encoder for Font Control to generate, create UPC Code image in Font applications. Data Matrix ECC200 Printer In None Using Barcode maker for Online Control to generate, create ECC200 image in Online applications. Matrix Barcode Creation In .NET Framework Using Barcode encoder for ASP.NET Control to generate, create Matrix 2D Barcode image in ASP.NET applications. Decode GS1128 In VB.NET Using Barcode scanner for Visual Studio .NET Control to read, scan read, scan image in Visual Studio .NET applications. 
