Continuing with some thoughts on helping students read math books, we will now look at the main things we find in them in addition to definitions which we discussed previously: theorems and axioms.
An implication is a sentence in the form IF (one or more things are true), THEN (something else is true). The IF part gives a list of requirements, so to speak, and when the requirements are all met we can be sure the THEN part is true. The fancy name for the IF part is hypothesis; the THEN part is called the conclusion.
Implications are sometimes referred to as conditional statements – the conclusion is true based on the conditions in the hypothesis.
An example from calculus: If a function is differentiable at a point, then it is continuous at that point. The hypothesis is “a function is differentiable at a point”, the conclusion is “the function is continuous at that point.”
This is often shortened to, “Differentiability implies continuity.” Many implications are shortened to make them easier to remember or just to make the English flow better. When students get a new idea in a shortened form, they should be sure to restate it so that the IF part and the THEN part are clear to them. Don’t let them skip this.
Related to any implication are three other implications. The 4 related implications are:
- The original implication: if p, then q.
- The converse is formed by interchanging the hypothesis and the conclusion of the original implication: if q, then p. Even if the implication is true, the converse may be either true or false. For example the converse of the example above, if a function is continuous then it is differentiable, is false.
- The inverse is formed by negating both the hypothesis and the conclusion: if p is false, then q is false. For our example: if a function is not differentiable, then it is not continuous. As with the converse, the inverse may be either true or false. The example is false.
- Finally, the contrapositive is formed by negating both the original hypothesis and conclusion and interchanging them, if q is false, then p is false. For our example the contrapositive is “If a function is not continuous at a point, then it is it is not differentiable there.” This is true, and it turns out a useful. One of the quickest ways of determining that a function is not differentiable is to show that it is not continuous. Another example is a theorem that say if an infinite series, an, converges, then . This is most often used in the contrapositive form when we find a series for which ; we immediately know that it does not converge (called the nth-term test for divergence).
The original statement and its contrapositive are both true or both false. Likewise, the converse and the inverse are both true or both false.
Any of the 4 types of statements could be taken as the original and the others renamed accordingly. For example, the original implication is the converse of the converse; the contrapositive of the inverse is the converse, and so on.
Definitions are implications for which the statement and its converse are both true. This is the real meaning of the reversibility of definitions. For this reason, definitions are sometimes called bi-conditional statements.
Axioms and Theorems
There are two kinds of if …, then… statements, axioms (also called assumptions or postulates) and theorems. Theorems can be proved to be true; axioms are assumed to be true without proof. A proof is a chain of reasoning starting from axioms, definitions, and/or previously proved theorems that convinces us that the theorem is true. (More on proof in a future post.)
It would be great if everything could be proved, but how can you prove the first few theorems? Thus mathematical reasoning starts with (a few carefully chosen) axioms, and accepts them as true without proof. Everything else should be proved. If you can prove it, it should not be an axiom.
Theorems abound. All of the important ideas, concepts, “laws” and formulas of calculus are theorems. You will probably see few, if any, axioms in a calculus book, since they came long before in the study of algebra and geometry.
When teaching students and helping them read and understand their textbook, it is important that they understand what a theorem is and how it works. They should understand what the hypothesis and conclusion are and how they relate to each other. They should understand how to check that the parts of the hypothesis are all true about the function or situation under consideration, before they can be sure the conclusion is true.
For the AP teachers this kind of thing is tested on the exams. See 2005 AB-5/BC-5 part d, or 2007 AB-3 parts a and b (which literally almost no one got correct). These questions can be used as models for making up your own questions of other theorems.