MATH220 December 2011
• Q1 (a) • Q1 (b) • Q1 (c) • Q1 (d) • Q1 (e) • Q1 (f) • Q2 (a) • Q2 (b) • Q3 • Q4 • Q5 (a) • Q5 (b) • Q6 (a) • Q6 (b) • Q7 (a) • Q7 (b) • Q8 •
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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Why doesn't the function
have an inverse?
What about the function
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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
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,
and so this really is the inverse of ƒ.
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