Science:Math Exam Resources/Courses/MATH220/December 2011/Question 01 (e)

MATH220 December 2011

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[hide] Question 01 (e)
Let A, B be non-empty sets, and let ƒ: A → B be a function. When does ƒ have an inverse function ƒ-1? Define ƒ-1.

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If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

[show] Hint
Why doesn't the function ${\begin{aligned}f:\{0,1\}&\to \{3\}\\0&\mapsto 3\\1&\mapsto 3\end{aligned}}$ have an inverse? What about the function ${\begin{aligned}g:\{0\}&\to \{1,2\}\\0&\mapsto 2\end{aligned}}$

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[show] Solution

A function ƒ has an inverse if and only if it is bijective. That is, it has an inverse if and only if it is Injective (or one-to-one) Surjective (or onto). In such a case, for every y ∈ B, there is a unique x ∈ A such that ƒ(x) = y. Using this fact, we define ƒ-1 by the rule $\displaystyle f^{-1}(y)=x$ where x is the unique element of the set A such that ƒ(x) = y. From the injectivity and surjectivity, this is well defined. Moreover, $f{\big (}f^{-1}(y){\big )}=f(x)=y$ and $f^{-1}{\big (}f(x){\big )}=f^{-1}(y)=x$ and so this really is the inverse of ƒ.

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