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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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