# injective and surjective

A function f from a set X to a set Y is injective (also called one-to-one) ant the other onw surj. The rst property we require is the notion of an injective function. Recall that a function is injective/one-to-one if . Some examples on proving/disproving a function is injective/surjective (CSCI 2824, Spring 2015) This page contains some examples that should help you finish Assignment 6. De nition. f(x) = 1/x is both injective (one-to-one) as well as surjective (onto) f : R to R f(x)=1/x , f(y)=1/y f(x) = f(y) 1/x = 1/y x=y Therefore 1/x is one to one function that is injective. If f is surjective and g is surjective, f(g(x)) is surjective Does also the other implication hold? It is injective (any pair of distinct elements of the domain is mapped to distinct images in the codomain). Theorem 4.2.5. It is also not surjective, because there is no preimage for the element $$3 \in B.$$ The relation is a function. Note that some elements of B may remain unmapped in an injective function. INJECTIVE, SURJECTIVE AND INVERTIBLE 3 Yes, Wanda has given us enough clues to recover the data. Then we get 0 @ 1 1 2 2 1 1 1 A b c = 0 @ 5 10 5 1 A 0 @ 1 1 0 0 0 0 1 A b c = 0 @ 5 0 0 1 A: Injective, Surjective and Bijective One-one function (Injection) A function f : A B is said to be a one-one function or an injection, if different elements of A have different images in B. The function is also surjective, because the codomain coincides with the range. (See also Section 4.3 of the textbook) Proving a function is injective. However, sometimes papers speaks about inverses of injective functions that are not necessarily surjective on the natural domain. a ≠ b ⇒ f(a) ≠ f(b) for all a, b ∈ A f(a) […] A function f: A -> B is said to be injective (also known as one-to-one) if no two elements of A map to the same element in B. Thus, f : A B is one-one. On the other hand, suppose Wanda said \My pets have 5 heads, 10 eyes and 5 tails." Thank you! Injective and surjective functions There are two types of special properties of functions which are important in many di erent mathematical theories, and which you may have seen. Injective (One-to-One) Determine if Injective (One to One) f(x)=1/x A function is said to be injective or one-to-one if every y-value has only one corresponding x-value. Injective and Surjective Functions. Let f(x)=y 1/x = y x = 1/y which is true in Real number. We also say that $$f$$ is a one-to-one correspondence. Formally, to have an inverse you have to be both injective and surjective. Hi, I know that if f is injective and g is injective, f(g(x)) is injective. I mean if f(g(x)) is injective then f and g are injective. surjective if its range (i.e., the set of values it actually takes) coincides with its codomain (i.e., the set of values it may potentially take); injective if it maps distinct elements of the domain into distinct elements of the codomain; bijective if it is both injective and surjective. 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