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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 \(f(4)\)?

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.

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 \(f(c)\)?

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 \(Y?\)

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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