Consider the subset S = {0, 2, 4, 6, 8, 10, 12} of Z14Z. (a) Show t.pdf

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Consider the subset S = {0, 2, 4, 6, 8, 10, 12} of Z/14Z. (a) Show that S = {0, 2, 4, 6, 8, 10, 12} forms a subring of Z/14Z. (b) Show that S forms a unital ring. (c) Show that S is not a unital subring of Z/14Z. Solution (a) Notice that all elements of S can be written as 2k Let 2k, 2j in S Then 2k+2j = 2(k+j) is in S, so clearly sum is in S Also (2k)(2j) = 4kj = 2(2kj) is in S and so product is also in S We can also see that 0, the additive identity of Z/14Z is in S by definition Finally if 2k is in S, then -2k is in S and so S is subring of Z/14Z (b) The ring has identity 8 Because 2.8 = 16=2, 4.8= 32 = 4, 6.8 = 48 = 6, 8.8 = 64 = 4, 10.8 = 80 = 10, 12.8 = 96 = 12 Hence S is unital (c) As we can see that we have {2,4,6,10,12} and here 8 is missing and so S is not unital subring of Z/14Z.

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C++ BinaryTree Help : Creating main function for Trees... Here are the header files : PrintExpressionTour.H: #ifndef _PRINTEXPRESSIONTOUR_H_ #define _PRINTEXPRESSIONTOUR_H_ #include \"EulerTour.h\" #include template class PrintExpressionTour : public EulerTour { protected: // ...same type name shortcuts as in EvaluateExpressionTour public: inline void execute( const BinaryTree& T ) { // execute the tour initialize( T ); std::cout << \"Expression: \"; eulerTour( T.root() ); std::cout << std::endl; } protected: // leaf: print value virtual void visitExternal( const Position& p, Result& r ) const { std::cout << *p; } // left: open new expression virtual void visitLeft( const Position& p, Result& r ) const { std::cout << \"(\"; } // below: print operator virtual void visitBelow( const Position& p, Result& r ) const { std::cout << *p; } // right: close expression virtual void visitRight( const BinaryTree::Position& p, Result& r ) const { std::cout << \")\"; } }; #endif // _PRINTEXPRESSIONTOUR_H_ BinaryTree.h: #ifndef _BINARYTREE_H_ #define _BINARYTREE_H_ #include #include typedef std::string Elem; // base element type class BinaryTree { protected: struct Node { // a node of the tree Elem elt; // element value Node* par; // parent Node* left; // left child Node* right; // right child Node() : elt(), par( NULL ), left( NULL ), right( NULL ) {} // constructor }; public: class Position { // position in the tree private: Node* v; // pointer to the node public: Position( Node* _v = NULL ) : v( _v ) {} // constructor Elem& operator*() const // get element { return v->elt; } Position left() const // get left child { return Position( v->left ); } Position right() const // get right child { return Position( v->right ); } Position parent() const // get parent { return Position( v->par ); } bool isRoot() const // root of the tree? { return v->par == NULL; } bool isExternal() const // an external node? { return v->left == NULL && v->right == NULL; } friend class BinaryTree; // give tree access }; typedef std::list PositionList; // list of positions public: BinaryTree(); // constructor int size() const; // number of nodes bool empty() const; // is tree empty? Position root() const; // get the root PositionList positions() const; // list of nodes void addRoot(); // add root to empty tree void expandExternal( const Position& p ); // expand external node Position removeAboveExternal( const Position& p ); // remove p and parent // housekeeping functions omitted... protected: // local utilities void preorder( Node* v, PositionList& pl ) const; // preorder utility private: Node* _root; // pointer to the root int n; // number of nodes }; #endif // _BINARYTREE_H_ EulerTour.h: #ifndef _EULERTOUR_H_ #define _EULERTOUR_H_ #include \"BinaryTree.h\" template // element and result types class EulerTour { // a template for Euler tour protected: struct Result { // stores tour results R leftResult; // result from left subtree R rightResult; // result from right subtree R finalR.

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Write a short essay explaining the four primary groups of production. Include in your explanation an example of each group as well as what we call the payment each group receives. Solution Four primary groups of production are four main factors of production which are a foundation of any productions. these four factors are land, labor, capital, and entrepreneurship. the first factor is Land that consists all type of natural resources used to create goods and services. examples : land, Gold, oil, forest, animal etc. Labor who transform these natural resources into usable resources. the value of labor depends on their skills and educations. to increase the productivity of firm owner trained their employees (labors) to become more skilled. capital includes buildings, tools, and machines that are labor usees to make outputs of the process. some examples are trucks, automated machines, computers, etc. entrepreneurship who create the idea and combine these all three factors of productions and makes a profit and increases the productivity of a country. payments to land and capital are rent, wages to labor, and profit to entrepreneurship..

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