×
Discrete Mathematics

# 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?

×