Domain and Range Properties

[Adrian Nicoara]

I do not understand the logic behind this property:
S: A <-> B
dom S ⊆ A
Why would the domain of S be taken as a subset of A??
This would be like defining
f:R->R, f(x)=sqrt(x)
Of course I can define the domain over R, but sqrt(x) is invalid for
negative values of x, if the range is R..., what would be the point of
that domain, if it is invalid for half of it's values??
The same goes for the range, except that the argument can be made with
f(x)=x^2.
In other words, the problem that I have is with the definition of a
relation over a domain or range that it doesn't fully cover.
Can you please elaborate on why a relation would be defined this way,
and not using minimal sets?
Thanks

[Nancy Day]

Post by Adrian Nicoara
S: A <-> B
dom S ⊆ A
Why would the domain of S be taken as a subset of A??
This would be like defining
f:R->R, f(x)=sqrt(x)
Of course I can define the domain over R, but sqrt(x) is invalid for
negative values of x, if the range is R..., what would be the point of
that domain, if it is invalid for half of it's values??
The same goes for the range, except that the argument can be made with
f(x)=x^2.

The type of a relation (which include functions) are the universal sets
and it is very common to have the dom R or ran R be subsets of the universal set.
For example, the type of the domain of a relation might be the set of students and the
actual domain of the relation will be a subset of those students. In a spec,
a relation might change its domain over the course of the system execution, but
it does not change its type.

For function, f, if the dom f not= type of the domain, it is a non-total
function.

cheers, nancy
CS245 Winter 2009 Instructor

Post by Adrian Nicoara
In other words, the problem that I have is with the definition of a
relation over a domain or range that it doesn't fully cover.
Can you please elaborate on why a relation would be defined this way,
and not using minimal sets?
Thanks