Talk:Intersection (set theory)

Jump to: navigation, search
WikiProject Mathematics (Rated Start-class, High-importance)
This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of Mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.
Mathematics rating:
 Start Class
 High Importance
Field: Foundations, logic, and set theory

Why are so many SQL implementations incomplete in so many aspects? Is there a reason for why INTERSECT ALL is not implemented in other versiond of SQL, such as performance, complexity or lack of demand that can be inserted into this article? Zaphraud 05:26, 30 May 2007 (UTC)

====================

I have the same problem comprehending intersection. To me it doesnt seem like a viable operation because once you put a "line"/marker in the middle of a box etc, the 2 small rectangles hence formed are seperate set.

so for example if I had a set of "all people in the school" and I wanted to make a set of all girls in the school I could use references to the people in the database, but then the "reference" itself is pointing to some memory cell etc, so I am actually making another set which contains refrences to a set of memory cells which have girls. Is that correct?

Likewise, if I have a set of all people who wear red ties to school, then I have another set whose elements are memory locations of the people who wear red ties to school.

Now if I want to find all the girls who wear red ties to school, then either I do search on the set of girls or the set of girls or the set of reds to find the matching.

The answer I get is a set of memory locations of the girls wearing red ties which I have to "store" in another memory location.

This set may contain elements of the set girl and the elements of the girl red, but however, this is a set seperate from the base set of girls and red ties. I mean one could also say that set contains ref to memory cells from the original set, but each of these references is always stored in a seperate set of cells..so it is indeed a seperate set.

Hence intersection cannot be a single step operation ever. It is like saying A.B is a subset of B, whereas in any digital circuit, one can tell you that A.B and B will have 2 seperate outcomes (they need 2 seperate LEDs to be percieved)

You can say B and B.A are the 2 events, however one would still say B is different from B.A , just like the domain name a.com is different from b.a.com

Note even if b belongs to a.com, it is a seperate entity and cannot be called a.com, it is called b.a.com —Preceding unsigned comment added by Alokdube (talkcontribs) 06:37, 30 June 2008 (UTC)

Please note the same logic applies when you consider a set of points on f(x,y)=0 and g(x,y)=0 The locus of points on their intersection is GF(x,y)=f(x,y)^2 + g(x,y)^2 =0 is different from each individual locus. That is precisely where the difference starts. If you have a window with a frame in it, the frame is different from the window, you will call it as window.frame only. It may inherit some properties from the parent window, but it is a seperate entity. The button click on the frame is an event on the frame. —Preceding unsigned comment added by Alokdube (talkcontribs) 06:53, 30 June 2008 (UTC)

The intersection of A and B is written "A ∩ B". Formally: x ∈ A ∩ B if and only if x ∈ A and x ∈ B.

so does this imply that if x ∈ C along with being an element of A and B, x is not ∈ A ∩ B —Preceding unsigned comment added by 192.18.192.21 (talk) 10:35, 3 August 2009 (UTC)

No JumpDiscont (talk) 03:15, 25 January 2010 (UTC)

Intersection of the empty set

The following definition of the intersection does result in an empty set for the empty set:

Let G(X) = {(z,y): y exists such that z in y and y in X}

Let the Intersection(X) be the set { u: v exists such that (u,v) in G(X) } (projection)

Now for X=0 (empty set) there is no such y in X and therefore G(X) is empty.

In the next step, since no such v can exist since G(X) is empty, then no u satisfies the condition and therefore the Intersection(0) = 0

Please point out what is wrong in this approach and give a set for which it doesn't work. I suspect it works, and you might come up with a set that contains elements for which it is undecideable whether the element belongs to the set. This IMHO would not affect the intersection as defined above, but to handle these, we would need a to define a set as a pair of X, Y: X={x can be decided to be in the set}, Y= {y can be decided to be not in the set}. Highlander (talk) 01:12, 2 May 2010 (UTC)

It looks like you have defined the union of an empty family, and that sort of union is indeed empty. For intersection, relative to some domain of discourse U, we would have
Let I(X) = {z : z in U and, for all y in X, z in y }
so, if X is empty, then I(X) is exactly U. — Carl (CBM · talk) 01:33, 2 May 2010 (UTC)
No, my definition above does intersection, at least for the set not being empty. But, well I rest my points, because I did confuse the intersection of the empty set of sets with the intersection of the set containing just the empty set. —Preceding unsigned comment added by Highlander (talkcontribs) 20:29, 3 May 2010 (UTC)
According to your definition, "Intersection(X)" is the set of all u such that there is some v with uv and v ∈ X. That is what is normally called the union of the set X. — Carl (CBM · talk) 00:38, 4 May 2010 (UTC)

Nullary intersection section seems odd to me too

The discussion in the article appears to be confusing a difficulty with finding the right way to specify n-ary intersection in formal logic with the usual understanding of the n-ary intersection operation.

The usual definition of arbitrary intersection of M (under ZF set theory) is the set containing all the common members of all the member sets of M.

• If M={} there are no members of members, so ⋂M = {}.
• If M has a single member {x} then ⋂M = x.
• If M has two members {x,y}, then ⋂M = x ∩ y.
• If M has three members {x, y, z}, then ⋂M = x ∩ y ∩ z
• and so on...

Specifying it in formal logic is a bit tricky. Suppose we say ⋂M := ∃S∀z(z ∈ S)∀y(y ∈ M)( ∀y( z ∈ y)). If M is empty, then this simplifies to ∃S∀z(z ∈ S)∀y(y ∈ M)(). This means that there are no constraints at all on S, i.e. the result of n-ary intersection. This is not what we want so we need a different way of saying it.

A better way is ⋂M := ∃S∀z∀y(y ∈ M)(z ∈ S ≡ ∃ y ∧ ∀y( z ∈ y)).

This forces S to be empty whenever M is empty since z being an element of S is equivalent to saying that M has at least one element. This fits much better with the normative understanding of the intersection of M. (see http://tedsider.org/teaching/st/st_notes.pdf ,p. 7)

77.126.146.103 (talk) 20:03, 4 December 2012 (UTC)

Confusion in Basic Definition section

The phrase "If the sets A and B are closed under complement..." is confusing. I looked up closure, this phrase says (i) if the complement of A is within A and (ii) if the complement of B is within B. I suspect that the writer meant that A and B are within universe U, and (i) the complement of A is within U and (ii) the complement of B is within U. From a nontheoretician, this seems obvious; it's hard for me to picture a case in which A, B, and U are defined in such a way that complementing A or B yields members outside of U.131.136.242.1 (talk) 22:44, 23 January 2013 (UTC)