Mathematical Functions
In mathematics, a function is an equation where you “plug in”
a value, and get an “answer” so to speak. In particular,
whenever you plug in a particular value, you must get a
SINGLE answer. (You should also get the same answer
always.) Functions graphed on the xy plane have to pass the
vertical line test.
Now, in discrete mathematics, we will be using functions a bit
differently & we will also coin a new term “relation”. In
particular, a function is a specific type of relation.
In standard high school mathematics, we typically deal with
functions of one variable. We always graph a function of the
form y=f(x), where the left hand side is entirely dependent on
x. Depending on what the function f(x) is, there is always a set
of values that are VALID to “plug” in to the equation.
This set
is the domain. Similarly, the “answer” you get out of the
function will always lie in a particular set. This set is the range.
The problem with using standard functions for discrete
mathematics is that many are defined for all real numbers.
Namely, it would be nice if we could list every value in the
domain of some function. But, we CAN NOT list out each real
number. (We can list out each integer however...)
The basis of functions and relations in discrete mathematics is
the idea that values of a domain and range should be subsets of
a set that can be listed, such as the integers, color, etc.
As we go through different things, I will make analogies to
mathematical functions, so you can see the similarities between
these and the functions and relations for discrete mathematics.
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Relations
A relation is something that relates one set of values to another
set of values. Sometimes the relationship that is specified
between sets is meaningful, other times it is not.
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 Fall '09
 Set Theory, Binary relation, Cartesian product, aRb, Composition of relations

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