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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.

# Bijective Functions

Let $$X$$ and $$Y$$ be the sets \begin{align} X &= \{ x \mid -12 \leq x \leq 4, x \text{ is an integer} \} \\ Y &= \{ y \mid 4 < y < a, y \text{ is an integer} \} \end{align} for some integer $$a$$. If there exists a bijective function from $$X$$ to $$Y$$, what is the value of $$a?$$

Consider the sets $X=\{a,b,c,d\}, Y=\{7, 13, 19,25\},$ and let $$f$$ be a bijective function from $$X$$ to $$Y$$. If $$M$$ and $$m$$ are the maximum and minimum values of $$f(a)+f(b)$$, respectively, what is $$M-m$$?

For two sets $X=\{x \mid -1 \leq x \leq 1\}, Y=\{y \mid 2 \leq y \leq 16\},$ a bijective function $$f: X \to Y$$ is defined as $$f(x)=ax+b$$, where $$a > 0$$. What is $$ab$$?

For set $$A=\{-5, 3, 8, 18\}$$, how many functions from $$A$$ to $$A$$ are not bijective?

For two sets $X=\{x\mid x\geq 11\}, Y=\{y\mid y\geq 121\},$ function $$f: X \to Y$$ is defined as $f(x)=x^2-2x+a.$ If $$f$$ is a one-to-one, bijective function, what is the value of the constant $$a$$?

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