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Section A.2 Reading Mathematics

To read mathematics is very different from reading a novel or a magazine article in that you need to pay extra attention to how terms are defined and used, as well as to the logical structure of the text. This is because mathematics is a precise language, and a word used in an English sentence may mean something different than when used in a proof, for instance.

Let's look at a short proof.

Let's prove the statement: An integer is even if and only if its cube is even.

Proof

(\(\Rightarrow\)) Let \(m\) be an integer. First, we show that \(m\) being even implies \(m^3\) is even. If \(m\) is even, then we can write \(m = 2k\) for some integer \(k\text{.}\) Then,

\begin{align*} m^3 \amp = (2k)^3\\ \amp = 8k^3\\ \amp = 2(4k^3)\text{.} \end{align*}

Since \(k\) is an integer, then \(4k^3\) must be an integer as well. This shows that \(m^3\) is even.

(\(\Leftarrow\)) Next, we show that \(m^3\) even implies \(m\) is even. To do this we prove the contrapositive: If \(m\) is odd, then \(m^3\) is odd.

If \(m\) is odd, then there is an integer \(k\) such that \(m = 2k + 1\text{.}\) Then

\begin{align*} m^3 \amp = (2k+1)^3\\ \amp = 8k^3 + 12k^2 + 6k + 1\\ \amp = 2(4k^3 + 6k^2 + 3k) + 1\text{.} \end{align*}

Since \(k \in \mathbb{Z}\text{,}\) then \(4k^3 + 6k^2 + 3k\) is an integer as well. Therefore \(m^3\) is odd.

This completes the proof.

After reading the proof in Example A.2.1 answer the following questions:

  1. What was/were being proven? What method(s) were used?
  2. Explain how the variable \(m\) is being used in the proof. What does it represent?
  3. Explain how the quantity \(k\) is being used in the proof. What does it represent?
  4. What definitions were used, either implicitly or explicitly?
  5. What were the key words/terms that helped you understand the structure of the proof?

For longer proofs involving multiple objects it is less straightforward to keep track of which ones represent arbitrary quantities, which ones have been fixed, and so on. Here are some general tips for reading math.

Remark A.2.3. Tips for Reading Mathematics.
  • Take notes: use pen and paper, or digital tools, to write down your thoughts as you read the proof. Draw diagrams, write an outline, emphasize key points, highlight confusing steps that need further study.
  • Look for key words to help you understand the logical structure of the argument: therefore, implies, since, assume, thus, and so on.
  • Mathematical proofs often rely on previous results, so you will frequently have to look up past propositions or theorems. You'll also notice that some results are more useful than others.

  • Speaking of propositions and theorems: it is not always clear what the difference is between them (and lemmas, properties, principles, and so on…).

    • Theorems are the most important results in a mathematical text or paper.
    • Propositions are results that are lesser in importance.
    • Lemmas are minor results that are specifically used to prove theorems or other results.
    • Corollaries are direct consequences (e.g. a special case) of theorems.
    • Conjectures are mathematical statements that are not yet proven, but they ‘make sense’, i.e. there is some consensus that they could be true. The Twin Prime Conjecture is a famous example.
    • Principles are similar to theorems, but are usually deemed to be more ‘fundamental’ in nature, or generally applicable.

  • Don't rush through the material. Ask yourself questions to check understanding, for example:

    • Are there any words whose meaning is vague or unclear to me?
    • Do I believe the proof?
    • Are there other arguments that could have worked?
    • Can I simplify the idea to make it more understandable?
    • Can I come up with my own examples based on this definition or property?
    • Why did the author choose this way of explaining it?
    • and so on…
  • Also think about the big picture. What were the assumptions? What was being proven? How does this proof and the methods used fit into the rest of the text?
  • Finally, try explaining it to someone else. Having discussions with other classmates will deepen your own understanding of the results (and will usually reveal whether one actually understands the material or not).

Let \(f: A \rightarrow B\) be a bijective function and assume it is strictly increasing. This means that for all \(x_1,x_2\) in \(A\text{,}\)

\begin{equation*} x_1 \lt x_2 \Rightarrow f(x_1) \lt f(x_2)\text{.} \end{equation*}

To show that \(f^{-1}\) is strictly increasing, we need to show that

\begin{equation*} \text{If } y_1 \lt y_2, \text{ then } f^{-1}(y_1) \lt f^{-1}(y_2) \end{equation*}

for all \(y_1,y_2 \in B\text{.}\)

We prove this by contradiction. Take \(y_1,y_2 \in B\) such that \(y_1 \lt y_2\text{,}\) and assume that \(f^{-1}(y_1) \geq f^{-1}(y_2)\text{.}\)

Note that we must have \(f^{-1}(y_1) > f^{-1}(y_2)\) since \(f^{-1}\) is injective.

Since \(f\) is strictly increasing, we have

\begin{align*} f^{-1}(y_1) \amp \gt f^{-1}(y_2) \\ \Rightarrow y_1 \amp \gt y_2 \end{align*}

contradicting the assumption that \(y_1 \lt y_2\text{.}\) Thus \(f^{-1}(y_1) \lt f^{-1}(y_2)\) must be true.

Go back to your MAT102 notes [3] and pick a theorem. Read this theorem, applying the tips in Remark A.2.3.