Math Problem Statement
Find a bijection g : [0, 1] −→ [0, 1), and make sure to prove it is a bijection.
Solution
To construct a bijection , we need a function that is both injective (one-to-one) and surjective (onto).
One possible bijective function is:
\begin{cases} x & \text{if } x \in [0, 1) \\ \frac{1}{2} & \text{if } x = 1. \end{cases}$$ ### Step-by-Step Analysis 1. **Domain and Codomain**: The function $$g$$ is defined for all $$x \in [0, 1]$$ and maps to values in $$[0, 1)$$. 2. **Definition of the Function**: - For $$x \in [0, 1)$$, $$g(x) = x$$. - For $$x = 1$$, $$g(1) = \frac{1}{2}$$. 3. **Injectivity**: - For $$x, y \in [0, 1)$$, $$g(x) = x$$ and $$g(y) = y$$. If $$g(x) = g(y)$$, then $$x = y$$, which preserves injectivity on $$[0, 1)$$. - For $$x = 1$$, $$g(1) = \frac{1}{2}$$, which does not overlap with any values $$g(x) = x$$ for $$x \in [0, 1)$$. - Therefore, $$g$$ is injective. 4. **Surjectivity**: - Every $$y \in [0, 1)$$ has a preimage in $$[0, 1]$$: for $$y \in [0, 1)$$, set $$x = y$$, and then $$g(x) = x = y$$. - Therefore, $$g$$ is surjective onto $$[0, 1)$$. Since $$g$$ is both injective and surjective, it is bijective. This function $$g$$ meets the requirements for a bijection between $$[0, 1]$$ and $$[0, 1)$$. Would you like a deeper explanation or have any questions? ### Related Questions 1. What are other examples of bijections between closed and half-open intervals? 2. How does the continuity of a function affect its bijectivity? 3. Can we define a bijection between $$(0, 1)$$ and $$[0, 1]$$? 4. How would you define a bijection from $$[0, 1]$$ to $$[a, b)$$ for any $$a < b$$? 5. Can we extend this function $$g$$ to work on all real numbers? **Tip:** A function is bijective if it is both one-to-one (injective) and onto (surjective). Always check both conditions when verifying a bijection.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Set Theory
Functions
Bijection
Injectivity
Surjectivity
Formulas
g(x) = x if x ∈ [0, 1); g(x) = 1/2 if x = 1
Theorems
Definition of Bijection
Properties of Injective and Surjective Functions
Suitable Grade Level
Undergraduate - Introduction to Higher Mathematics
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