China 
Africa 
Explore chapters and articles related to this topic
Vector analysis
Published in John P. D’Angelo, Linear and Complex Analysis for Applications, 2017
Recall that a function f : A → B is injective if f(a) = f(t) implies a = t. An injective function has an inverse: for y in the image of f, we define f−1(y) to be the unique a with f(a) = y. A function f : A → B is surjective (or onto) if the image of A under f is all of B. Suppose f is both injective and surjective. By surjectivity, for each b ∈ B there is an a ∈ A with f(a) = b. By injectivity, this a is unique. This surjectivity guarantees existence of a solution and injectivity guarantees uniqueness of the solution.
Preliminary Content
Published in John D. Ross, Kendall C. Richards, Introductory Analysis, 2020
John D. Ross, Kendall C. Richards
Loosely speaking, a function is injective if there are never two points in the domain that map to the same point in the codomain. An extreme example of a function that is not injective is the constant function f:R→R given by f(x) = 1, since we can find two points in the domain (say, x = 0 and x = 17), that both map to the same point in the codomain (f(0) = f(17) = 1).
Combinatorics
Published in Paul L. Goethals, Natalie M. Scala, Daniel T. Bennett, Mathematics in Cyber Research, 2022
Let r,s∈ℙ, and let R,S be r- and s-element sets. Recall that a function is injective (or one-to-one) if each element of the domain maps to a unique element of the range, surjective (or onto) if every element of the range has a preimage, and bijective if it is both injective and surjective.
Mathematical understanding and ownership in learning: affordances of and student views on templates for proof-writing
Published in International Journal of Mathematical Education in Science and Technology, 2022
Sarah Klanderman, V. Rani Satyam
Furthermore, proof frameworks provide structure for when intuition and formal proof differ. Using the example of set equality, Selden and Selden (2007) discussed how set inclusion and the method of ‘element-chasing’ to prove set equality are mentally different. The concept image (Tall & Vinner, 1981) associated with set inclusion, where one set is contained inside another, may feel different to the student from picking an arbitrary element in one set and showing it is also in another. Another example is a one-to-one (injective) function, where every output in a domain has only one associated input. To the student, understanding the intuitive meaning of an injective function versus rigorously proving a function is injective based on its definition, by supposing that f(x1) = f(x2) and then deducing that x1 = x2, can feel like different processes.