Logical structure of definitions
Roughly speaking, there are two types of Math: formula Math, where all calculations can be seen, and abstract (or invisible) Math, which happens in the head. In my Optimization class last year, students had serious problems with abstract Math. The problems started at the definitions level, which meant the students couldn’t move any further. This post is my attempt to help them with abstract Math. There is established terminology that people use to describe rules of logic. Good luck with that terminology. Below I introduce my own. Trying to explain something at an elementary level is always a thankless task, so I apologize in advance for lapses.
Math uses the usual human logic
Example 1. In each subject, our students have to take two exams: one at KBTU and another (at a later date) at the University of London (UoL). Experience shows that the UoL grade on average is 30 points lower than mine. The passing grade at the UoL is 40. Therefore I call my student successful if he/she gets at least 70 points (out of 100) in my class.
Each definition has a preamble, which provides logical grounds for the definition. All objects under consideration are elements of one large set, which I call an encompassing set. The set we are defining is a subset of the encompassing set. We use a certain property to separate elements in the set we are defining from those which are not its elements. This defining property should make sense in the encompassing set.
In Example 1, the preamble is “In each subject … passing grade at the UoL is 40.” The encompassing set is “Students taking my class”, the set we are defining is “Successful students”. The defining property is “Getting at least 70 in my class”. This property wouldn’t make sense if we used “All KZ citizens” as the encompassing set.
Let denote the encompassing set, the set we are defining and its complement. The original definition (of ) is called a direct definition. The definition of is its opposite. The direct definition and its opposite should be stated in such a way that and a) do not intersect and b) together cover the whole encompassing set. In proofs by contradiction often it is necessary to formulate the opposite definition.
The set can be defined using two types of definitions. In the first type we directly negate the property that is used to define . I call such a definition a negative opposite. In the second type we use the opposite of the property . I call such a definition a positive opposite. The opposite of “Getting at least 70 in my class” is “Getting less than 70 in my class”.
Example 2. I call a student unsuccessful if he/she does not get a grade of at least 70 in my class (negative opposite). I call a student unsuccessful if he/she gets a grade lower than 70 in my class (positive opposite).
In the negative opposite we don’t change the defining property; we simply say that it is not satisfied. In the positive opposite we pass from the property to its opposite and say that the elements satisfy the opposite property. The negative opposite contains negation “does not”. The positive opposite does not contain it (it may contain negations in more complex cases).
... but applies it to unusual objects
One and the same definition can be formulated in several different (however, equivalent) ways. Always try to find the most geometric form and then establish equivalence of different definitions.
Example 3. A set is called bounded from above if there is a number such that is a subset of a half-infinite interval : . Any such is called an upper bound for .
I start with this version of the definition because it is the most visual. Note that we require existence of a number with a certain property. A slightly shorter version of the above definition is:
A set is called bounded from above if is a subset of a half-infinite interval , for some .
In this version, the word “some” indicates existence. When we say “for all”, we affirm universality. A replacement of an existence requirement by a universal requirement drastically changes the definition. See what we get from Example 3 by such a replacement.
Example 4. A set is called bounded from above if for any number , is a subset of a half-infinite interval . Equivalently, a set is called bounded from above if is a subset of a half-infinite interval , for any .
Can you tell for which this definition holds? Some people can answer this question without hesitation. If you are not one of them, try to move the number . Many mathematical arguments require a choice of an object that would allow the researcher to prove or disprove a statement. In Example 4, we say “for any M”, and it is UP TO YOU to try any and select the ones which show what is going on.
Example 5. An inclusion relation in terms of sets equivalently in terms of set elements takes the form for all . If we replace the set relation by the element-wise relation in Example 3, we obtain a longer definition: a set is called bounded from above if there is a number such that for all .
This is the place people start having problems. The errors that I saw include confusion of “there is” (existence) and “for all” (universality). See what is wrong with the following three definitions (what does not correspond to the intuition of being “bounded from above”; we also don't want to be empty).
Example 6. A set is called bounded from above if for any number there exists such that .
Example 7. A set is called bounded from above if for any number and for any one has .
Example 8. A set is called bounded from above if there is such that for any number one has .
Example 9. A set is called bounded from above if for any there exists a number such that .
Remark. In Example 6, first is chosen and depends on it. In Example 7, the choices of and are independent. In Example 8, is fixed and is arbitrary. Finally, in Example 9, first is chosen and the choice of depends on .
In the next exercises, start with the most graphic version. In case of doubt, do everything step by step (describe the encompassing set, defining property etc.).
Exercise 1. Define a set bounded from below using a) sets and b) element-wise terminology. In both cases formulate the opposite definition and the definition of a lower bound.
Exercise 2. Define a bounded set (that is, bounded from below and above) using a) sets and b) element-wise terminology. In both cases formulate the opposite definition.