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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.

Definition

         

\(15\) football teams are competing in a knock-out tournament. Each game has a winner, there are no draws, and the losing team is out of the tournament. How many games need to be played in order for a tournament champion to be determined?

For two sets \[X=\{-1,1,a\}, Y=\{8,9, b\},\] \(f(x)=x^3+9\) is a bijective function from \(X\) to \(Y\). What is the value of \(a+b\)?

For the two sets \(X=\{a, b, c\}\) and \(Y=\{9, 17, 72\},\) how many bijective functions are there from \(X\) to \(Y\)?

Suppose \(f(x)=ax+3\) is a bijective function from \(\mathbb{R}\) to \(\mathbb{R}\). If \(f(7)=38,\) what value of \(x\) satisfies \(f(x) =103?\)

Suppose \(f(x)=ax+3\) is a bijective function from \(\mathbb{R}\) to \(\mathbb{R}\). If \(f(5)=28,\) what value of \(x\) satisfies \(f(x) =78?\)

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