Are set notations simplifyable?

[lioric]

TL;DR Summary

I tried to name the shaded area of a Venn diagram using numbers to isolate the regions. And I found that there are several ways to get the same region.
Can the set notations simplfy

I tried to name the shaded area of a Venn diagram using numbers to isolate the regions. And I found that there are several ways to get the same region.
Can the set notations simplfy

 

[Mark44]

Summary:: I tried to name the shaded area of a Venn diagram using numbers to isolate the regions. And I found that there are several ways to get the same region.
Can the set notations simplfy

I tried to name the shaded area of a Venn diagram using numbers to isolate the regions. And I found that there are several ways to get the same region.
Can the set notations simplfy

What notations do you mean? Your question is very unclear. If the shaded region is common to sets A and B, the notation $A \cap B$ represents the intersection of the two sets. See https://en.wikipedia.org/wiki/Set_theory.

 

[lioric]

What notations do you mean? Your question is very unclear. If the shaded region is common to sets A and B, the notation $A \cap B$ represents the intersection of the two sets. See https://en.wikipedia.org/wiki/Set_theory.

What I mean is for instance $A \cap B$ is equal to $(A' \cup B')'$
It indicates the same area.
Is there a way to simply this. Like the way fractions get simplified or algebra gets simplified

 

[SSequence]

What I mean is for instance $A \cap B$ is equal to $(A' \cup B')'$
It indicates the same area.
Is there a way to simply this. Like the way fractions get simplified or algebra gets simplified

Yes, essentially one can use much of the apparatus of boolean algebra/logic to describe what you are talking about. So because "intersection" corresponds to "+" and union corresponds to "." ... they should be enough to describe any possible "areas of intersection" for arbitrary finite number of sets n with A1,A2,A3,...,An.

But several other operations should be enough too.

 

[FactChecker]

There are many identities within set theory that can give you different representations of the same area. I believe that the smallest parts of the Venn diagram are traditionally referred to as the conjunction (connected by 'and') of the main sets A, B, C,... , A', B', C',...

 

[SSequence]

To expand a bit on the last post if we had five "atomic propositions" $a,b,c,d,e$ then an expression like:
$(a\cdot b \cdot c \cdot d \cdot e)+(a \cdot b \cdot c \cdot d \cdot e')+(a \cdot b' \cdot c' \cdot d \cdot e)$
would change to [replacing the lower-case letter case letters to upper-case to highlight them as sets]:
$(A\cap B \cap C \cap D \cap E) \cup (A \cap B \cap C \cap D \cap E') \cup (A \cap B' \cap C' \cap D \cap E)$

And, for example, any identity has a counter-part in propositional algebra (and vice versa). For example:
$A \cap B= (A' \cup B')'$
changes to following equivalent in propositional expression/equation:
$a \cdot b= (a' + b')'$

 

[mfb]

There are many ways to simplify these expressions, and there is not always a unique shortest way to describe a set. The one from the post above can be simplified to:
$$(A\cap B \cap C \cap D) \cup (A \cap B' \cap C' \cap D \cap E)$$
using the rule
$$(X \cap Y) \cup (X \cap Y') = X$$

 

[lioric]

There are many ways to simplify these expressions, and there is not always a unique shortest way to describe a set. The one from the post above can be simplified to:
$$(A\cap B \cap C \cap D) \cup (A \cap B' \cap C' \cap D \cap E)$$
using the rule
$$(X \cap Y) \cup (X \cap Y') = X$$

Could you tell me which topic of mathematics teaches these. Or subtopics
I would like to brush up on them
Thank you very much

 

[Mark44]

What I mean is for instance $A \cap B$ is equal to $(A' \cup B')'$
It indicates the same area.
Is there a way to simply this. Like the way fractions get simplified or algebra gets simplified

Yes. Look up De Morgan's Laws -- https://en.wikipedia.org/wiki/De_Morgan's_laws

 

[FactChecker]

After learning about DeMorgan's Laws, you might want to take a look at some of the Normal Forms of boolean logic, which are very closely related. They give some standard ways of expressing sets. Look at Conjunctive Normal Form (CNF) and Disjunctive Normal Form (DNF)

 

[sysprog]

You would be opening some boxes: Cantor, Zermelo, Frankel, Gödel, Russell, Euler, Boole -- searching on any of those names would lead to a sea of set-theory results -- I think that you would do well to look at Cantor's diagonal argument en route to getting a feel for the territory.

 

[lioric]

Thank you all for your help