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

MATH220 December 2011

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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.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

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!

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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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. 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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