Probability

Bijections

Injection and Surjection

         

Consider the two sets X={1,2,3,4},Y={8,0,20}.X=\{1,2,3,4\}, Y=\{-8, 0, 20\}. Let ff be a surjective function from XX to YY such that for any two elements x1x_1 and x2x_2 of X,X, if x1<x2,x_1 < x_2, then f(x1)f(x2).f(x_1) \leq f(x_2). What is the minimum possible value of f(4)f(4)?

A function ff maps the elements of A={14,16,18,20}A = \{14, 16, 18, 20\} to elements of B={55,66,77,88,99}.B =\{55, 66, 77, 88, 99\}. How many of the possible maps ff 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},X=\{a,b,c\}, Y=\{7, 11, 13, 17, 25, 32\}, ff is an injective function from XX to YY. If f(a)=7f(a)=7 and f(b)=17f(b)=17, what is the sum of all the elements of YY that can possibly be the value of f(c)f(c)?

For X={2,1,0,1,2},X=\{-2,-1,0,1,2\}, function ff is surjective from XX to Y,Y, where f(x)={x+4 if x>0,x3+18 if x0.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?Y?

For two sets X={a,b,c},Y={y1y7,y is an integer},X=\{a,b,c\}, Y=\{y\mid 1 \leq y \leq 7, y \text{ is an integer}\}, how many injective functions f:XYf: X \to Y exist?

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