Try to discover the definition of a function (from domain X to codomain Y), denoted f:X->Y,...

 

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Try to discover the definition of a function (from domain X to codomain Y), denoted f:X->Y, on your own in the following manner: • Discover a definition of the graph of a function f:X->Y. • Prove the graph of a function f:X->Y is a relation between X and Y. • Formulate "the vertical line test" in this parlance, i.e. state what it means for a vertical line to intersect a relation R between X and Y, i.e. a subset R CXXY more than once in terms a criterion regarding the elements (x, y) of the relation R. o It might help to think of what goes wrong for geometric objects (e.g. in the plane) to render them not a function, i.e. y not a function of x. For example, why is the circle, i.e. the set of solutions (x, y) to x² + y² = 12 not such that y is a function of x, and so not the graph of any function of x? o What about the sideways parabola, i.e. the set of solutions (x, y) to y² = x? Using your new formulation of "the vertical line test" for relations, define a relation R between X and Y, i.e. a subset RC X x Y, to be a function f:X->Y iff the graph of f satisfies "the vertical line test". ● Try to discover the definition of a function (from domain X to codomain Y), denoted f:X->Y, on your own in the following manner: • Discover a definition of the graph of a function f:X->Y. • Prove the graph of a function f:X->Y is a relation between X and Y. • Formulate "the vertical line test" in this parlance, i.e. state what it means for a vertical line to intersect a relation R between X and Y, i.e. a subset R CXXY more than once in terms a criterion regarding the elements (x, y) of the relation R. o It might help to think of what goes wrong for geometric objects (e.g. in the plane) to render them not a function, i.e. y not a function of x. For example, why is the circle, i.e. the set of solutions (x, y) to x² + y² = 12 not such that y is a function of x, and so not the graph of any function of x? o What about the sideways parabola, i.e. the set of solutions (x, y) to y² = x? Using your new formulation of "the vertical line test" for relations, define a relation R between X and Y, i.e. a subset RC X x Y, to be a function f:X->Y iff the graph of f satisfies "the vertical line test". ●

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To discover a definition of the graph of a function f X Y we can start by considering the Cartesian product of X and Y denoted as X x Y The graph of t... View the full answer