Math notes / 'elementary mathematics' and similar concepts

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This is more for overview of my own than for teaching or exercise.

Overview of the areas

Arithmetic · 'elementary mathematics' and similar concepts
Set theory, Category theory
Geometry and its relatives · Topology
Elementary algebra - Linear algebra - Abstract algebra
Calculus and analysis
Logic
Semi-sorted
 : Information theory · Number theory · Decision theory, game theory · Recreational mathematics · Dynamical systems · Unsorted or hard to sort


Math on data:

  • Statistics as a field
some introduction · areas of statistics
types of data · on random variables, distributions
Virtues and shortcomings of...
on sampling · probability
glossary · references, unsorted
Footnotes on various analyses

Other data analysis, data summarization, learning

Regression · Classification, clustering, decisions · dimensionality reduction · Optimization theory, control theory
Connectionism, neural nets · Evolutionary computing



Terms like elementary mathematics (and various others) often means something like

  • "things common to many mathematical fields"
  • "generally useful, taught once comfortable with arithmetic"


Much of it is less than day-to-day, but useful with some frequency.

Ideally, these are the things that turn up with some frequency in non-math fields, for example in various academic fields, and in particular those related to physics and engineering.


Common number sets

  • ℤ - whole numbers, positive and negative integers, {...,-2,-1,0,1,2,...}
  • ℕ - natural numbers, used in real-world counting.
    • Whether zero is included in this depends a little who you ask - which field of math (or computer scientists). To be unambiguous, it is now relatively common to use:
    • 0 to refer to {0,1,2,...} and
    • * or ℕ1 to refer to {1,2,3,...}
(Use of a plus is discouraged as confusing, as in ℤ+ and ℝ+ it signifies non-negative)
  • ℚ - rational numbers, those in the form a/b where a∈ℤ and b∈ℤ but b≠0.
    • You also see ℚ+ and ℚ- as non-negative and non-positive rationals
  • ℝ - real numbers.
    • You also see ℝ+ referring to non-negative reals (including 0), and ℝ- as non-positive reals (including 0)

Note that ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ (you can bake a subset typology of these and quite a few more, see e.g. number set venn diagram)

Superscript tends to refer to multiple-dimensional vectors. For example vectors in ℝ2, for example (1,2)


Complex set


See also:

Functions/maps, images, and such

While the intuitive use of these is fairly obvious, it is useful, sometimes crucial, to understand the more formal definitions as well.

Practical functions often deal with (infinite) number sets, though in example diagrams (also for injection/bijection/surjection) you may see small finite sets.


Given A and B as two sets, a function or map maps elements in A to B. More specifically, each element in A is associated with an element in B. For each such pair, the element in B is the image of the respective element in A (sometimes called the argument).


A function in math is much like a function as programmers might be used to: You hand in a value, name it x, and get back a value, f(x). Exactly how that happened is only your concern if you make it that.


In such a relation, A is called the domain (the 'input' for which the function is defined), B is the range (and sometimes image).


('Map' and 'function' are usually synonyms of each other. People tend to have preference for one or another, or may e.g. prefer 'map' when using with smallish finite sets and 'function' when dealing with a named general-purpose mappers.)


You can describe a formula

  • by a formula (in purer mathematics often specifically using set theory)
  • by its relationship to other functions - for example in the case of inverse functions
  • by an algorithm that computes it
  • by a description of its properties
  • by table, plot or graph (often in applied contexts)


On names

Common enough functions have widely used names (e.g. sin, abs, and such) with well-settled definitions.

It is also common to name functions (often letters like f, g), when they are used more than once in an analysis of a problem - or just out of this habit.

You will also see use of mostly f purely within an otherwise anonymous function definitions.

When a name isn't needed at all, such as in one-off mentions, you will also see nameless function definitions (such as y=2x+1 instead of f(x)=2x+1).


Notations related to (set theory style) function definitions include:

  • f : A → B (function details to be mentioned later)
  • f : ℝ → ℝ with f = x ↦ ex
  • f : ℝ → ℝ+ : x ↦ ex


You could define functions for specific domain/range intervals, for example when you want to avoid certain problem cases outside a range that applies to a specific problem (though many prefer to work with functions that are as general as possible) for example:

  • f : [-1,1] → [0,1] : x ↦ something


Elements on which functions act need not be atomic. That is, you can define things as being members of the cartesian product of sets. For example:

  • f : ℝ×ℝ → ℝ might define some flattening from two-dimensional coordinates to one, ro so.


(functions of multiple arguments can however be a little more involved, with more combinations that can lead to problem cases)


An/the identity function (completely) maps a/any set to itself with identical values.


mapping properties:

This article/section is a stub — probably a pile of half-sorted notes, is not well-checked so may have incorrect bits. (Feel free to ignore, fix, or tell me)

While there are more, injective, surjective, and bijective are most interesting for most fields of mathematics.


Maps/functions (for the sake of brevity, assume a map from A to B) are...

  • injections / injective functions are those where each image is mapped to by at most one (so zero or one) argument.
    • Intuitively, that mappings are one-to-one, though not necessarily using all the values of B.
    • Applying this function and then its inverse would be the identity function on


  • surjections / surjective functions are those where every image is mapped to by one (or more) arguments -- when for all b∈B, there is at least one a∈A so that f(a)=b.


  • bijections/bijective for things that are injective and surjective. In other words, when the mapping maps all values in one set to all values in the other in a one-to-one style, and leaving no unmapped elements in either set. This is in effect a permutation (although that is harder to untuit for non-finite sets).


These properties can be useful in some restrictions and definitions. For example, a function is bijective iff it has an inverse. Intuitively speaking, for each value in B you can


http://en.wikipedia.org/wiki/Bijection,_injection_and_surjection


See also the properties of functions heading under abstract algebra.

Logarithms

The logarithm of a number, to a given base, is the power to which the base must be raised to produce the number. For example, the logarithm of 100 in base 10, or log101000 is 3 (103 = 1000).

http://en.wikipedia.org/wiki/Logarithms


The most commonly used bases are:

  • e (natural logarithm)
  • 10
  • 2 (happens where binary choices are involved, e.g. in computer algorithmics)

Mathematicians and computer scientists have been a little inconsistent in implying different bases, meaning that when you see log(x), you may have to deduce or guess.

ISO, NIST, and others suggest that to avoid confusion, you should use ln to indicate e, lg for base 10, and lb for base 2. I've not much seen the last, but the other two seem to be pretty well accepted.