That's backwards. In a Venn diagram relations are the big circle and functions are the smaller circle inside it. A function is a deterministic relation.
If you mean to say a table is a function, then yes. All relations, including functions, can be represented using a table.
If a row is listed more than once (e.g. the row 3 9) it still doesn't affect the definition of my function. 3 still maps to 9.
This is the issue really. I think what you're saying is:
{1,2,3,3} == {1,2,3}
That's true mathematically, but that's not what's happening in SQL. Try creating the table to represent the square function. Now try inserting (3,9) into it twice. SQL won't let you because it forces tables to represent relations. Not queries though.
In my example {1,2,3} is the direct representation of the value of the set and {1,2,3,3} is a different representation. The tables and queries in SQL are direct representations. If a SQL query looks like |1|2|3|3| then what you actually have, mathematically, is {{1,2,3},{3}}, aka multi-set or bag.
{{1,2,3},{3}} != {1,2,3}
That's what's happening when a row is listed more than once. It does affect the definition of your function.
> That's backwards. In a Venn diagram relations are the big circle and functions are the smaller circle inside it. A function is a deterministic relation.
That's partially right. In a Venn diagram, whenever you see a little circle S contained in a big circle T what you have is a relation that says S is a subset of T. Specifically this is a relation that maps a set of sets to {TRUE, FALSE}. Fundamentally we have sets and mappings[1] between sets. A relation is a mapping between some set and a set of logical values. So relations are mappings but not all mappings are relations. In the case of SQL a relation is a well-defined mapping so we call it a 'function'.
> If a SQL query looks like |1|2|3|3| then what you actually have, mathematically, is {{1,2,3},{3}}, aka multi-set or bag.
No sir. What you have, formally, is a programmatically generated relation/function (under a closed-world assumption) in which one of the entries in the mapping has been overly specified. Your relation maps the set {1,2,3, 4 ...} to the set {TRUE, FALSE}. If you pass it the value 1 or 2 or 3 you get TRUE. If you pass it the value 4 you get FALSE. The domain of this function is {1,2,3, 4, ...} and the codomain (or range) is {TRUE, FALSE}. Having an entry listed twice in the definition is similar to the following:
f(x) = 0 when x = 0 and x * x if x is real.
That function has been overly specified. Now if we had a table with entries like this:
1 Fred Jones
2 James Smith
3 Jane Doe
4 Alice Baker
3 George Washington
then we have a problem since the 'function' is now ambiguous (and hence not a function).
Now, back to my employee table example. If I have a table called Employees defined as:
where EMP_ID is the primary key then what I have is a composition of functions (a relation in SQL is always a function)
One function, we'll call it G maps full names to Integers and has this 'signature': S x S -> Z where S is the set of strings and Z is the set of integers.
The other function, we'll call it H maps Integers to {TRUE, FALSE} so my table is actually equivalent to the function H of G.
Now I realize that this is all abstract and a bit pedantic but it is nonetheless what SQL and relational algebra are all about even if we don't ordinarily think of it like this.
[1] Formally, mappings, functions, transformations and operations are all the same things just used in specific contexts.
That's backwards. In a Venn diagram relations are the big circle and functions are the smaller circle inside it. A function is a deterministic relation.
If you mean to say a table is a function, then yes. All relations, including functions, can be represented using a table.
If a row is listed more than once (e.g. the row 3 9) it still doesn't affect the definition of my function. 3 still maps to 9.
This is the issue really. I think what you're saying is:
That's true mathematically, but that's not what's happening in SQL. Try creating the table to represent the square function. Now try inserting (3,9) into it twice. SQL won't let you because it forces tables to represent relations. Not queries though.In my example {1,2,3} is the direct representation of the value of the set and {1,2,3,3} is a different representation. The tables and queries in SQL are direct representations. If a SQL query looks like |1|2|3|3| then what you actually have, mathematically, is {{1,2,3},{3}}, aka multi-set or bag.
That's what's happening when a row is listed more than once. It does affect the definition of your function.