To define a bijection $g: [0,1] \to [0,1)$, we need a function that is both injective (one-to-one) and surjective (onto). The function must map each element in the closed interval $[0,1]$ to a unique element in the half-open interval $[0,1)$, and every element in $[0,1)$ must have a preimage in $[0,1]$.

Step 1: Define the function

Let's define the following function:

\begin{cases} \frac{x}{2}, & \text{if } x \in [0, 1) \\ \frac{1}{2}, & \text{if } x = 1 \end{cases}$$ - For all $$x \in [0,1)$$, the function maps $$x$$ to $$\frac{x}{2}$$, which stays in the interval $$[0,1)$$. - At $$x = 1$$, we map it to $$\frac{1}{2}$$, which is also in the interval $$[0,1)$$. ### Step 2: Prove that $$g$$ is injective (one-to-one) To prove injectivity, we need to show that if $$g(x_1) = g(x_2)$$, then $$x_1 = x_2$$. - For $$x_1, x_2 \in [0,1)$$, if $$g(x_1) = g(x_2)$$, this means $$\frac{x_1}{2} = \frac{x_2}{2}$$. Hence, $$x_1 = x_2$$. - For $$x_1 = 1$$, the only $$x$$ that maps to $$\frac{1}{2}$$ is 1 itself by definition, so no other element in $$[0,1)$$ can map to $$\frac{1}{2}$$. Thus, $$g$$ is injective. ### Step 3: Prove that $$g$$ is surjective (onto) To prove surjectivity, we need to show that for every $$y \in [0,1)$$, there exists an $$x \in [0,1]$$ such that $$g(x) = y$$. - For $$y \in [0,1/2)$$, we can take $$x = 2y$$. Since $$y \in [0,1)$$, the corresponding $$x \in [0,1)$$, and $$g(x) = \frac{x}{2} = y$$. - For $$y = \frac{1}{2}$$, we have $$g(1) = \frac{1}{2}$$, so there exists $$x = 1$$ that maps to $$\frac{1}{2}$$. Thus, $$g$$ is surjective. ### Conclusion The function $$g(x) = \frac{x}{2}$$ for $$x \in [0,1)$$, and $$g(1) = \frac{1}{2}$$, is both injective and surjective, so it is a bijection from $$[0,1]$$ to $$[0,1)$$. --- Would you like further details or have any questions? Here are five related questions to explore further: 1. Can you construct another bijection from $$[0,1]$$ to $$[0,1)$$? 2. How would the proof change if you wanted to find a bijection from $$(0,1]$$ to $$(0,1)$$? 3. Can you find a bijection between $$[0,1]$$ and $$(0,1)$$? 4. How would you extend the concept of a bijection to higher dimensions, e.g., $$[0,1]^n$$? 5. What are examples of bijections between unbounded intervals, such as $$[0,\infty)$$ and $$(0,\infty)$$? **Tip**: When proving injectivity, remember to focus on showing that different inputs map to different outputs. For surjectivity, ensure every output has a corresponding input.