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Bijections

Bijections, surjections, and injections are three types of functions which associate the elements between two sets. For example, each word in this sentence can be mapped to exactly one in the last.

Injection and Surjection

         

Consider the two sets \[X=\{1,2,3,4\}, Y=\{-8, 0, 20\}.\] Let \(f\) be a surjective function from \(X\) to \(Y\) such that for any two elements \(x_1\) and \(x_2\) of \(X,\) if \(x_1 < x_2,\) then \(f(x_1) \leq f(x_2).\) What is the minimum possible value of \(f(4)\)?

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A function \(f\) maps the elements of \(A = \{14, 16, 18, 20\}\) to elements of \(B =\{55, 66, 77, 88, 99\}.\) How many of the possible maps \(f\) are not injective?

Details and assumptions

A function is injective if each element in the codomain is mapped onto by at most one element in the domain.

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For two sets \[X=\{a,b,c\}, Y=\{7, 11, 13, 17, 25, 32\},\] \(f\) is an injective function from \(X\) to \(Y\). If \(f(a)=7\) and \(f(b)=17\), what is the sum of all the elements of \(Y\) that can possibly be the value of \(f(c)\)?

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For \(X=\{-2,-1,0,1,2\},\) function \(f\) is surjective from \(X\) to \(Y,\) where \[f(x) = \begin{cases} x+4 & \text{ if } x > 0, \\ -x^3+18 & \text{ if } x \leq 0. \end{cases} \] What is the sum of all the elements of \(Y?\)

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For two sets \[X=\{a,b,c\}, Y=\{y\mid 1 \leq y \leq 7, y \text{ is an integer}\},\] how many injective functions \(f: X \to Y\) exist?

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