Exercises and Proofs

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Exercises and Proofs#

There are two extensions for jupyterbook that enable us to add exercises and proofs in our lecture series.

These extensions provide a beautiful and clear separation from the text and offer a better reading experience.

Exercises#

See also

To learn more about sphinx-exercise please read the documentation.

Here we provide a brief tutorial on using exercise admonitions in the quantecon lectures

To make an exercise you can use the exercise admonition

```{exercise}
:label: my-exercise

Recall that $n!$ is read as "$n$ factorial" and defined as
$n! = n \times (n - 1) \times \cdots \times 2 \times 1$.

There are functions to compute this in various modules, but let's
write our own version as an exercise.

In particular, write a function `factorial` such that `factorial(n)` returns $n!$
for any positive integer $n$.
```

where the label enables you to reference the exercise using ref or numref roles.

This will be rendered as

Exercise 1

Recall that \(n!\) is read as “\(n\) factorial” and defined as \(n! = n \times (n - 1) \times \cdots \times 2 \times 1\).

There are functions to compute this in various modules, but let’s write our own version as an exercise.

In particular, write a function factorial such that factorial(n) returns \(n!\) for any positive integer \(n\).

and then you can easily reference the exercise using the {ref} role (Exercise 1) or using the {numref} role (Exercise 1).

See also

There are additional configuration options available but are not frequently used in QuantEcon lectures

Each exercise admonition should be paired with a solution admonition.

The exercise extension also provides us with a useful property that enables us to convert our solutions into a dropdown paragraph.

This helps the reader to have more time to think about the questions without being tempted by the solution.

You can add this feature in class.

Guiding Principle

  • Use gated syntax whenever your exercise uses:

    • executable code cells

    • any nested directives (such as math, note etc.)

  • Use :class: dropdown for solutions by default

```{solution-start} my-exercise
:class: dropdown
:label: my-solution
```

Here's one solution.

```{code-cell} python
def factorial(n):
    k = 1
    for i in range(n):
        k = k * (i + 1)
    return k

factorial(4)
```

```{solution-end}
```

which is rendered as

The solution can be referenced using the label property using ref and numref.

The solution directive requires the label of the exercise.

Given many solutions across QuantEcon projects typically require code execution we frequently make use of the gated syntax.

Note: If a solution does not require code execution it is possible to use the primary solution admonition.

Warning

The code in this example is not executed but is syntax highlighted

````{solution} my-exercise
:label: my-solution

Here's one solution.

```{code-block} python
def factorial(n):
    k = 1
    for i in range(n):
        k = k * (i + 1)
    return k

factorial(4)
```
````

which is rendered as

Solution to

Here’s one solution.

def factorial(n):
    k = 1
    for i in range(n):
        k = k * (i + 1)
    return k

factorial(4)

Proofs#

There is a range of directives provided by sphinx-proof:

  1. prf:proof

  2. prf:theorem

  3. prf:axiom

  4. prf:lemma

  5. prf:definition

  6. prf:criteria

  7. prf:remark

  8. prf:conjecture

  9. prf:corollary

  10. prf:algorithm

  11. prf:example

  12. prf:property

  13. prf:observation

  14. prf:proposition

  15. prf:assumption

Note

The prf: included in each directive name is required in both the directive and when referencing the admonition such as {prf:ref}. These belong to the Sphinx prf domain that supports references and linking.

An example proof:

Tip

This package does not support the gated syntax provided by sphinx-exercise, therefore you need to use nested ticks if including other directives such as math as is shown in this example.

````{prf:proof}
We'll omit the full proof.

But we will prove sufficiency of the asserted conditions.

To this end, let $y \in \mathbb R^n$ and let $S$ be a linear subspace of $\mathbb R^n$.

Let $\hat y$ be a vector in $\mathbb R^n$ such that $\hat y \in S$ and $y - \hat y \perp S$.

Let $z$ be any other point in $S$ and use the fact that $S$ is a linear subspace to deduce

```{math}
\| y - z \|^2
= \| (y - \hat y) + (\hat y - z) \|^2
= \| y - \hat y \|^2  + \| \hat y - z  \|^2
```

Hence $\| y - z \| \geq \| y - \hat y \|$, which completes the proof.
```
````

will render as

Proof. We’ll omit the full proof.

But we will prove sufficiency of the asserted conditions.

To this end, let \(y \in \mathbb R^n\) and let \(S\) be a linear subspace of \(\mathbb R^n\).

Let \(\hat y\) be a vector in \(\mathbb R^n\) such that \(\hat y \in S\) and \(y - \hat y \perp S\).

Let \(z\) be any other point in \(S\) and use the fact that \(S\) is a linear subspace to deduce

\[\| y - z \|^2 = \| (y - \hat y) + (\hat y - z) \|^2 = \| y - \hat y \|^2 + \| \hat y - z \|^2\]

Hence \(\| y - z \| \geq \| y - \hat y \|\), which completes the proof.

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