x is a real number since sums and quotients (except for division by 0) of real numbers are real numbers. If such a real number x exists, then 5x -2 = y and x = (y + 2)/5. (We need to show that x in R such that f(x) = y.) Proving or Disproving That Functions Are OntoĮxample: Define f : R R by the rule f(x) = 5x - 2 for all x R. A function is not onto if some element of the co-domain has no arrow pointing to it. In arrow diagram representations, a function is onto if each element of the co-domain has an arrow pointing to it from some element of the domain. f is called onto or surjective if, and only if, all elements in B can find some elements in A with the property that y = f(x), where y B and x A.Ĭonversely, a function f: A B is not onto y in B such that x A, f(x) y. Onto Functions Let f: A B be a function from a set A to a set B. Hence h(n 1) = h(n 2) but n 1 n 2, and therefore h is not one-to-one. Prove that h is not one-to-one by giving a counter example. On the other hand, to prove a function that is not one-to-one, a counter example has to be given.Įxample: Define h: R R is defined by the rule h(n) = 2n 2. Proof: Suppose x 1 and x 2 are real numbers such that f(x 1) = f(x 2). To prove a function is one-to-one, the method of direct proof is generally used.
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