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Section A.4 Writing Mathematics, Part II

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Okay, it's your turn to try writing a clear, concise proof of a statement! As we have seen, the use of English words makes proofs more approachable and understandable. Here's a small sample of commonly-used phrases in mathematical proofs. Note the use of the plural we instead of the singular I.

Declare intentions

  • We will prove...
  • We want to show that...
  • In order to prove... we...
  • At this point we need to find...
  • We consider the following cases...

Clarify implications

  • Since... then...
  • Because..., we have...
  • Therefore/thus/hence...
  • This means that...
  • The previous statement implies...

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Explain steps

  • By assumption, we know that...
  • By simplication/manipulation,rearranging,...
  • Because of property/theorem/definition, we have...

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Moreover—and this is something that takes some practice to get used to—mathematical formulas and equations, when used in English sentences, need to be treated as part of the grammar. A complete sentence is one that expresses a complete idea. For example, TJ baked brownies. is a complete sentence, while baked brownies is not. (The latter is called a phrase, or a fragment.)

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Activity A.4.1. Complete sentence or not?

For each statement below, decide whether it is a complete sentence or not.

  1. x^2.
  2. x^2 = 9.
  3. Because A⊆B.
  4. 6 ways of rearranging the letters of the word DOG.
  5. We will manipulate (nk).
  6. −1∈N.

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Something else to note is that the two-column proof method of writing statement and reason beside each other is not a best practice. Besides the fact that it is not really used by math practitioners, these are just lists of bullet points that sometimes even obfuscate the logical links among the parts of a solution. They are sometimes useful, like for organizing your thoughts, but generally it is better form to write in complete sentences.

Show that if \(a \gt b \gt 0\text{,}\) then \(a^2 \gt b^2\text{.}\)

A Two-Column Proof

Statement Reason
\(a \gt b\) Given
\(a - b \gt 0\) Add \(-b\) to both sides
\(a + b \gt 0\) Since \(a,b \gt 0\)
\(a^2 - b^2 \gt 0\) Multiply two previous inequalities
\(a^2 \gt b^2\) Add \(b^2\) to both sides

A Proof in Prose

Let \(a\) and \(b\) be positive real numbers with \(a \gt b\text{.}\) Then \(a-b\) and \(a+b\) are both positive. Hence their product \((a-b)(a+b) = a^2 - b^2\) is positive as well, implying \(a^2 \gt b^2\text{.}\)

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Activity A.4.2.

Write a clear and complete solution to the following exercise:

Count the number of nonnegative integer solutions to the equation

x1+x2+x3+x4=35

that satisfy x1≥4, x2≥1, and x3≤5.

Try solving it yourself first before looking at the hint!

Hint

Count the number of solutions that satisfy \(x_3 \geq 6\) first.