What is the definition of discrete mathematics?
Most math classes deal with the study and manipulation of functions over a
continuous domain; calculus and algebra for example examine functions
usually over the domain of all real numbers.
The branch of discrete mathematics focuses on problems that are not over a
continuous domain. For example, is it possible to visit 3 islands in a
river with 6 bridges without crossing any bridge more than once? That is a
discrete math problem (because there are a finite (fixed, discrete) number
of bridges). Or, what is the smallest number of telephone lines needed to
connect 200 cities? The numbers can be large and the logic can be complex,
but these type of problems are different from finding an optimal value for a
function where the domain can be 3, 3.14, 3.14159, or any real value.
Todd Clark, Office of Science
Discrete mathematics is a branch of mathematics -- actually many
branches -- that deals with quantities that only have discrete values. One
example is the integers: 0, 1, 2, 3, 4, 5, ..., Another example is the
same as the previous one but including the negative numbers
..., -5, -4, -3, -2, -1, 0, +1, +2, +3,... Here the notation ",...,"
means "and so on".
Another important branch of discrete mathematics are the polygons that
can be drawn inside a circle so that the points of the polygon lie on the
circumference of the circle. These polygons can have 3 or more sides and can
be convex, like triangles, squares, rectangles, pentagons,..., or they can
be convex like a five pointed star or a six pointed star. Such geometric
figures have the property that they can be rotated a certain number of
degrees and after doing so the result is an identical geometric figure. For
example a square can be rotated 0, 90, 180, 270, 360 degrees and if your
eyes were closed when this was done you couldn't tell whether or not the
square had (or had not) been rotated by those amounts. The mathematical
properties that follow from such operations form a category of mathematics
called "group theory".
Yet another branch of discrete mathematics (a very important one) are
the prime integers -- integers that are divisible only by themselves and by
"1". The first few prime numbers are: 2,3,5,7,11,13,17,19,... Prime numbers
have other properties that are unique to them, and pose many interesting
mathematical questions, many of which the answers are not known. It is
known, and can be proven, that there are infinitely many prime numbers.
That is, there is no largest prime number. However, a much more difficult
question where the answer is not known is: If I choose a certain number, say
1000, or 1,000,000 or 1,000,000,000 how many prime numbers are there whose
value is less than that certain number.
So you can see that the subject of "discrete mathematics" involves many
areas of mathematics.
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Update: June 2012