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Section4A Short Note on Proofs

Abstract mathematics is different from other sciences. In laboratory sciences such as chemistry and physics, scientists perform experiments to discover new principles and verify theories. Although mathematics is often motivated by physical experimentation or by computer simulations, it is made rigorous through the use of logical arguments. In studying abstract mathematics, we take what is called an axiomatic approach; that is, we take a collection of objects \(\mathcal S\) and assume some rules about their structure. These rules are called axioms. Using the axioms for \(\mathcal S\text{,}\) we wish to derive other information about \(\mathcal S\) by using logical arguments. We require that our axioms be consistent; that is, they should not contradict one another. We also demand that there not be too many axioms. If a system of axioms is too restrictive, there will be few examples of the mathematical structure.

A statement in logic or mathematics is an assertion that is either true or false. Consider the following examples:

  • \(3 + 56 - 13 + 8/2 \text{.}\)

  • All cats are black.

  • \(2 + 3 = 5\text{.}\)

  • \(2x = 6\) exactly when \(x = 4\text{.}\)

  • If \(ax^2 + bx + c = 0\) and \(a \neq 0\text{,}\) then

    \begin{equation*} x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. \end{equation*}
  • \(x^3 - 4x^2 + 5 x - 6\text{.}\)

All but the first and last examples are statements, and must be either true or false.

A mathematical proof is nothing more than a convincing argument about the accuracy of a statement. Such an argument should contain enough detail to convince the audience; for instance, we can see that the statement “\(2x = 6\) exactly when \(x = 4\)” is false by evaluating \(2 \cdot 4\) and noting that \(6 \neq 8\text{,}\) an argument that would satisfy anyone. Of course, audiences may vary widely: proofs can be addressed to another student, to a professor, or to the reader of a text. If more detail than needed is presented in the proof, then the explanation will be either long-winded or poorly written. If too much detail is omitted, then the proof may not be convincing. Again it is important to keep the audience in mind. High school students require much more detail than do graduate students. A good rule of thumb for an argument in an introductory abstract algebra course is that it should be written to convince one's peers, whether those peers be other students or other readers of the text.

Let us examine different types of statements. A statement could be as simple as “\(10/5 = 2\text{;}\)” however, mathematicians are usually interested in more complex statements such as “If \(p\text{,}\) then \(q\text{,}\)” where \(p\) and \(q\) are both statements. If certain statements are known or assumed to be true, we wish to know what we can say about other statements. Here \(p\) is called the hypothesis and \(q\) is known as the conclusion. Consider the following statement: If \(ax^2 + bx + c = 0\) and \(a \neq 0\text{,}\) then

\begin{equation*} x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. \end{equation*}

The hypothesis is \(ax^2 + bx + c = 0\) and \(a \neq 0\text{;}\) the conclusion is

\begin{equation*} x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. \end{equation*}

Notice that the statement says nothing about whether or not the hypothesis is true. However, if this entire statement is true and we can show that \(ax^2 + bx + c = 0\) with \(a \neq 0\) is true, then the conclusion must be true. A proof of this statement might simply be a series of equations:

\begin{align*} ax^2 + bx + c & = 0\\ x^2 + \frac{b}{a}x & = - \frac{c}{a}\\ x^2 + \frac{b}{a}x + \left( \frac{b}{2a} \right)^2 & = \left( \frac{b}{2a} \right)^2 - \frac{c}{a}\\ \left(x + \frac{b}{2a} \right)^2 & = \frac{b^2 - 4ac}{4a^2}\\ x + \frac{b}{2a} & = \frac{ \pm \sqrt{ b^2 -4ac}}{2a}\\ x & = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. \end{align*}

If we can prove a statement true, then that statement is called a proposition. A proposition of major importance is called a theorem. Sometimes instead of proving a theorem or proposition all at once, we break the proof down into modules; that is, we prove several supporting propositions, which are called lemmas, and use the results of these propositions to prove the main result. If we can prove a proposition or a theorem, we will often, with very little effort, be able to derive other related propositions called corollaries.