25 Functions

25 Functions - Handout #25 April 21, 2010 CS103 Robert...

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Handout #25 CS103 April 21, 2010 Robert Plummer Functions As used in ordinary language, the word function indicates dependence of a varying quantity on another. If I tell you that your grade in this class is a function of your overall average, you interpret this to mean that I have a rule for translating a number in the range of 0 to 100 into a letter grade. More generally, suppose two sets of objects are given: set A and set B; and suppose that with each element of A there is associated a particular element of B. These three things: the two sets and the correspondence between elements comprise a function. Here is a first definition, based on this idea of mapping: Definition A function f is a mapping from a set D to a set T with the property that for each element d in D, f maps d to a single element of T, denoted f(d). Here D is called the domain of f, and T is called the target or co-domain . We write this as f: D T. We also say that f(d) is the image of d under f, and we call the set of all images the range R of f. A mapping might fail to be a function if it is not defined at every element of the domain, or if it maps an element of the domain to two or more elements in the range: Consider the mappings shown in the diagram above. We note that examples a and b are both functions since every element in the first set (the domain) maps to a single element in the second set (the co-domain or target). Note that it is fine for two elements in the domain to map to the same element in the co-domain (as is the case in b ). We point out that example c is not a function, since there is an element in the domain which does not map to any element in the co-domain. Also, example d is not a function since there is an element in the domain that maps to more than one element in the co-domain. One way to define a function is to provide a table that shows the mapping for each element of the domain. For example, in a small class we might have the set of students S = {Maria, Clara. Tom, Dick, Harry} the set of possible grades G = {A, B, C, D, NP} a b c d
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2 One possible function f: S G would be: d f(d) Maria A Clara B Tom C Dick A Harry B The table completely defines the function by showing every mapping. Note that all the possible grades are not used, i.e., the range is not the same as the co-domain. It is still convenient to call this a function from S to G in order to indicate the possibilities. The following table does not define a function from S to G, because not every member of S is mapped to a grade: d Maria A Clara B Tom C Dick Harry B The following table does not define a function from S to G, because one member of S is mapped to two grades: d Maria A Clara B Tom C Dick A Harry B Harry NP Another way to define a function to specify a rule for how the function operates, rather than listing out the mapping. For example, using the common notations N: the set of natural numbers {1, 2, 3, . ..} Z: the set of all integers {. .., -2, -1, 0, 1, 2, . ..} we could define a function f : N
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25 Functions - Handout #25 April 21, 2010 CS103 Robert...

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