Regarding so-called "union compatibility",.suppose relations A and B
with equal headings, say named h. Suppose non-empty relation U with
only one attribute that is not in any other heading.

value i): A Join B projected on h, D&D style =
(A <AND> B) {h}

value ii): A Union B (with unrestricted <OR> and Union restricted to
equal headings) =
{A <OR> B)

Every tuple in A and every tuple in B is in value i), no tuple in A or B
is not in value i), likewise for value ii).

We could use the expression of value i) to "insert" to relvars, even
define a union view. So what makes union different from projection of join?

paul c wrote:
Oops again, sorry, that expression should be (A <AND> B <AND> U) {h}.

"paul c" <toledobythesea (AT) oohay (DOT) ac> wrote

If A and B have the same heading, then you're wrong. Suppose that no tuple
in A is also in B, then A Join B would be empty, as would A <AND> B <AND> U.
A Union B, on the other hand, would contain every tuple in either A or B.

The difference is as plain as the difference between logical disjunction and
conjunction.

Suppose that you have two relations with the same heading,
A{x,y} and B{x,y}, such that

A =
{{x:=1,y:=2),{x:=3,y:=3},{x:=5, y:=4}}

B =
{{x:=2,y:=2),{x:=3,y:=3},{x:=4, y:=4}}

Then

A JOIN B =
{{x:=3,y:=3}}

A UNION B =
{{x:=1,y:=2),{x:=2,y:=2),{x:=3,y:=3},{x:=4, y:=4},{x:=5, y:=4}}

Notice that the join only contains those tuples that appear on BOTH A and B,
but the union contains those tuples that appear in EITHER A or B.

Mr. Scott wrote:
Right you are, thanks. I was wondering about <rename> but left that op
out. .