Onto (Surjective) FunctionsDate: 11/03/98 at 03:47:31 From: Misaki Taro Subject: Onto (Surjective) function According to some texts, the "onto" function is defined in the following way: A function f from A to B is called onto if for all b in B there is an a in A such that f(a) = b. Such functions are also called surjective. I don't know why most of the definition of an onto function is defined in an "inverted" way. Why is "for all b in B" mentioned first, while "onto function f from A to B" is the very center of the definition? Is it necessary to state the range first, not the domain? To be honest, I don't quite understand this kind of function. Would you please explain this family of functions to me in a more detailed and illustrative way? Thanks a lot. Date: 11/03/98 at 05:57:12 From: Doctor Allan Subject: Re: Onto (Surjective) function Hi Misaki, Let me give you a specific example: Consider the function f(x) = 3*x and the sets: A = {2,3,9,12} B = {6,9,27,36} and C = {6,9,21,27,36} The function f from A to B is onto because no matter which element you take in B, you can find an element in A such that 3 times that element equals the element from B. Take as an example 27 in B. The element 9 in A equals 27 when multiplied by 3, and it works this way for all elements in B since: 2*3 = 6, 3*3 = 9, 9*3 = 27, and 12*3 = 36 But when we let the function f go from A to C it is not onto. That's because the element 21 in C can't be paired with one in A - there is no element in A such that 3 times that element equals 21. Hope this helps! Sincerely, - Doctor Allan, The Math Forum http://mathforum.org/dr.math/ |
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