A ∩ B ⊂ A: The intersection of A and B contains only elements common to both. Every such element is necessarily in A, so it is a subset of A.

A ⊂ A ∪ B: The union of A and B contains all elements of A (and possibly more from B), so A is always a subset of A ∪ B.

Combining them: A ∩ B ⊂ A ⊂ A ∪ B is a valid logical progression.

>>16793196
Draw a venn diagram.

>>16793198
So it reads like (((A ∩ B ) ⊂ A ) ⊂ A) ∪ B. ??
>>16793202
I don't know what that is and it doesn't appear on my notes.

>>16793205
https://en.wikipedia.org/wiki/Venn_diagram

Are ∩, ⊂ and ∪ at the same resolution order?

Maybe a better question is, how do you read A ∩ B ⊂ A ⊂ A ∪ B exactly, like how do you put it into words?

>>16793198
So they are 2 properties, not 1?

>>16793217
The (intersection of A and B) is contained in A, which is itself contained in the (union of A and B).

>>16793221
So unions and intersections have priority over inclusions?
So it reads like this?: (A ∩ B) ⊂ A ⊂ (A ∪ B)

>>16793224
Yes, that's right

>>16793226
Thank you anon.
How was I supposed to know that btw? I was reading it from left to right, and I was confused why it had no parentheses or why A⊂ A ⊂ A was put in twice, (what's the point of saying it includes itself). I'm studying this career again since a long time of not touching math but I don't remember this being taught anywhere.

>>16793219
dunno, dawg. I just threw everything on ChatGPT. GPT-5 knows more than >>>/sci/

>>16793231
Some things and conventions just aren't spelled out and you have to figure out how to interpret them from the context.

>>16793205
it reads
[A ∩ B] ⊂ A
A ⊂ [A ∪ B]

kind of like how a < b < c is
a < b
b < c

>>16793239
Welp, that sucks. It all makes sense now but I wasted half an hour because I didn't know the vocabulary in the first place and I assumed they had the same hierarchy.

Do these symbols have any kind of mathematical term?
⊂ seems to be a comparison symbol (it doesn't create content, it compares) and ∩ ∪ seem to be generators maybe? (They create content).

In the same sense as the + - and the > < symbols, I don't know how to explain it.

>>16793256
The inclusion is an example of a relation https://en.wikipedia.org/wiki/Relation_(mathematics)
Intersection and union are examples of operations (on sets)
Similarly, < and > are relations on numbers. +, - are operations on numbers.

>>16793196
wtf has this board turned into

>>16793269
Do relations and operations share a common concept or definition? Like, symbols, signs? How do you call all of those, and is there anything else to it, like dunno, boolean maybe ?

>>16793196
If you can read it you understand it. The conclusion is that you can't actually read set notation

>>16793205
if you wanna do brackets do this:
(A ∩ B ) ⊂ A ⊂ (A ∪ B)

File: D U N C.jpg (90 KB, 1050x450)

90 KB JPG

Probabaly part of what is giving you a hard time is that a bunch of the symbols are exactly the same, or look similar, just with rotation (pic related, as a humorous example). The stuff about intersection, union and subsets have already been pointed out, so I'd like to draw attention to another angle: the number of symbols, and the TYPE of each symbol.

We have nine symbols in a row, some of which are used with repetition. As the FPBP said, the upside-down U is intersection, and the right-side up U is union. these are binary operators in set theory, and the things they are operating on are two sets, A and B, which are the operands. We also have the sideways-U symbol, which denotes that one thing is a subset of the other. As a very crude analogy that doesn't exactly correspond to the OP, consider the following:

3 - 1 < 3 < 3 + 1

Here, we have another nine symbols (some repeated), andwe have two binary operations, plus and minus. The complete statement can be interpreted like this: "It is the case that 3 - 1 is less than 3, AND it is the case that 3 is less than 3 + 1". In this case we had basic arithmetic as opposed to set theory.

What's the key here? Binary operations, operands and RELATIONS are all three different things, denoted by different symbols. Ask yourself this question: is the expression " 2 + 2 " true, or false? I would suggest neither. However, " 2 + 2 = 4", a string of five symbols, is a RELATION that can be judged to be true. In this case, the SUBSET symbol is the RELATION symbol, the point at which we can judge a complete statement to be true or false. Now, the statement can be read:

"It is the case that the intersection of the sets A and B is a subset of A, AND it is ALSO the case that A is a subset of the union of A and B." That's what the OP says, it's making two truth claims (using the same relation symbol, again, frequently the whole point of relations is that they can be judged true or false), and then compounding them.

This leads us to one other very important point: bracketing, order of operations. We know from arithmetic and internet memes that establishing precedence with parentheses or other clear understanding is very important, and yet the OP also seems to be hung up on this. The OP expression is daringly elegant, dispensing with them altogether. The reason why this is possible is again because the operations are not the same thing as the RELATIONS. Bracketing can be useful to visually organize information (or to aid cognition), but in this case there is no ambiguity, for the same reason that there is no ambiguity in

3 - 1 < 3 < 3 + 1

We have three expressions, and two relation comparisons between the expressions. Writing

(3 - 1) < 3 < (3 +1)

is superfluous in this case, unless you want it as a kind of visual aid.

I was taught that ⊂ is strictly used for a proper subset (i.e., subset but not equal to, where a line is added to the bottom to refer to like 'subset or equal to'), but I've seen people use this same symbol for that as well. What is the correct definition for ⊂?

>>16793217
Given set A

its union with any set B Is
superset of A

Its intersection with any set B is
Subset of A

>>16793196
It literally spells AN B CACA UB, can't you read?