Is this a purely linguistic coincidence or is there some logical structure to this in Boolean Algebra?

[Metronome]

The set of all elements in Set A or Set B = Set A and Set B.

I may have learned this a long time ago, but I've been tripped up many times in further subjects on when to write intersections VS unions based on this equivalence (I think I'm finally cured now but this idea came back into my head).

Is there a logic behind this or = and equivalence, or is it just an artifact of English? I know it's not De Morgan, but I can't recall if anything else related or with and in this way.

 

[lev888]

The set of all elements in Set A or Set B

What does this mean?

 

[Dr.Peterson]

The set of all elements in Set A or Set B = Set A and Set B.

I may have learned this a long time ago, but I've been tripped up many times in further subjects on when to write intersections VS unions based on this equivalence (I think I'm finally cured now but this idea came back into my head).

Is there a logic behind this or = and equivalence, or is it just an artifact of English? I know it's not De Morgan, but I can't recall if anything else related or with and in this way.

I think what you may mean (which is not at all clear!) is that the union of sets A and B, [imath]A\cup B[/imath], is defined as the set of all items that are elements of A or elements of B; and students are often confused, thinking of this as "putting sets A and B together into one set". This is not a proper statement in math, but a faulty, informal way to think of it, that you need to rid yourself of. It is not an equivalence, but an error.

As I see it, the error arises from focusing on the entire set rather than individual elements. When use the word "or" here, we mean each individual element of A U B is an element of A or of B (or both). We don't mean that we can choose either the (entire) set A or the set B. Similarly, when we talk about the intersection [imath]A\cap B[/imath], we want all items that are (individually) elements of both A and B. This doesn't mean that we put both entire sets together (which is the union).

This can cause trouble when you are translating a word problem into set-theoretical terms; but it should not be a problem in doing the work.

Am I understanding your question correctly? If not, please explain in different words.

 

[Metronome]

Yes, I think you've interpreted my statement rightly!

...the union of sets A and B...is defined as the set of all items that are elements of A or elements of B; and students are often confused, thinking of this as "putting sets A and B together into one set". This is not a proper statement in math, but a faulty, informal way to think of it...

...when we talk about the intersection...we want all items that are (individually) elements of both A and B. This doesn't mean that we put both entire sets together (which is the union).

These sound like two contradictory statements to me (that unions do not mean putting sets together and that unions do mean putting sets together).

Surely thinking about putting sets A and B together into one set would be a useful concept we would want Boolean Algebra/set theory to capture in some way, no? Why doesn't the union capture this on its own, and what modification would make it fit?

 

[Dr.Peterson]

These sound like two contradictory statements to me (that unions do not mean putting sets together and that unions do mean putting sets together).

My point was that "putting sets together" is not a formal description, and does not mean "and". It's misleading.

And the statements you appear to see as contradictory were talking about different things!

• About union: "students are often confused, thinking of this as "putting sets A and B together into one set"."
  • Here I said that students are right in saying the union puts two sets together; but they are wrong in thinking of this as what we mean by "and".
• About intersection: "This doesn't mean that we put both entire sets together (which is the union)."
  • Here I said that the word "and" used of intersection doesn't mean what union means. I said, again, that it is the union that combines sets.

The union (where we use the word "or") combines two sets into one, but we don't use the word "and" there in speaking formally. The intersection (where we do use the word "and") does not. How is it a contradiction to say that one thing is what the other is not??

Surely thinking about putting sets A and B together into one set would be a useful concept we would want Boolean Algebra/set theory to capture in some way, no? Why doesn't the union capture this on its own, and what modification would make it fit?

... and that's what the union means. How did I say anything different? Just don't let the word "and" confuse you, by using it of a union.

What needs modification is your first sentence,

The set of all elements in Set A or Set B = Set A and Set B.

That is utter nonsense. If you use words correctly, you won't say such things. And if someone actually taught you that "equivalence", as you imply, then they should have been fired. At best, you could change it to this, taking the weight off the word "and" and putting it on other words:

The set of all elements in Set A or Set B, called the union, is the result of combining Set A and Set B into a single set.

A similar issue comes up in teaching students to translate words into algebraic expressions. In the phrase "the product of A and B", the word "and" does not imply addition; it just joins two entities that are to be multiplied. The emphasis should not be on "and", but on "product". Students who are taught too mechanically to translate words into operations get it wrong.

 

[Metronome]

Okay, I see where you're coming from now.

To me it is just extremely intuitive to say that, i.e., the union of y = x with y = 2x is y = x and y = 2x. In the multiplication example, the satisfying (inaccessible to young children) explanation is that "and" delimits the individual inputs of the product operation, identically to addition or any other binary operation (product of A and B means product(A, B), sum of A and B mean sum(A, B), blubber of A and B means blubber(A, B)).

I wonder if there is a satisfying explanation of this intuition in the form of some system of logic (logic of parts and wholes?) which equates or'ing over elements to and'ing over sets.

 

[Dr.Peterson]

To me it is just extremely intuitive to say that, i.e., the union of y = x with y = 2x is y = x and y = 2x.

The trouble here is that the union of the SETS {x,y | y=x} and {x,y | y=2x} is the set {x,y | y=x OR y=2x}; apart from (0,0), both equations can't be true, so "and" would be wrong.

You want to just say "the equations y = x and y = 2x together describe the union", but you have to say HOW they do so.

A classic situation where this arises is absolute value inequalities. The inequality |x| < 3 is equivalent to x < 3 AND x > -3; both must be true. (We can also write this as -3 < x < 3, in which both parts again must be true.) On the other hand, the inequality |x| > 3 is equivalent to x > 3 OR x < -3; both can never be true for the same x! Only one of the inequalities can be true at a time. This is a union: The solution set is the union of {x | x > 3} and {x | x < -3}. And it would be wrong to say that in this case x > 3 AND x < -3. (It would also be wrong to write -3 > x > 3; that would imply that -3 > 3!)

In the multiplication example, the satisfying (inaccessible to young children) explanation is that "and" delimits the individual inputs of the product operation, identically to addition or any other binary operation (product of A and B means product(A, B), sum of A and B mean sum(A, B), blubber of A and B means blubber(A, B)).

Yes, the "and" in "product of A and B" plays the role of a mere comma!

 

[Metronome]

[some random ramblings to my future self]Even if I define [math]\{x, y\ |\ y = x\} \cup \{x, y\ |\ y = 2x\} = \{x, y\ |\ y = x\}\ AND\ \{x, y\ |\ y = 2x\} = \{x, y\ |\ y = x\ OR\ y = 2x\}[/math] associating the union with [imath]AND[/imath] w.r.t. sets and [imath]OR[/imath] w.r.t. set elements, I cannot, per intuition, symmetrically define [math]\{x, y\ |\ y = x\} \cap \{x, y\ |\ y = 2x\} = \{x, y\ |\ y = x\}\ OR\ \{x, y\ |\ y = 2x\} = \{x, y\ |\ y = x\ AND\ y = 2x\}[/math]
The intersection is associated with [imath]AND[/imath] w.r.t. set elements, but there does not seem to be a word besides intersect itself to associate w.r.t. sets.

On the other side of the coin, some other set operation associated with [imath]OR[/imath] w.r.t. sets would appear to invoke a bit of quantum indeterminacy. It would not be symmetric with [imath]AND[/imath] w.r.t. sets to just have three copies of [imath]\mathbb{R}^2[/imath], each with one of [imath]y = x[/imath], [imath]y = 2x[/imath], and [imath]y = x \cup y = 2x[/imath], as that would represent a family of sets, which would not correspond to the sense in which the union is associated with [imath]AND[/imath] w.r.t sets.