Binary operation in algebraic structure

Algebraic Structure: Are Set Operations Considered Binary Operations? - Mathematics Stack Exchange

I would like to figure out how these things are related. I know I have a lot of reading to do which will probably take a lot of time.

Thankfully I found a helpful thread that I found I would like to recommend to others: Algebraic structure cheat sheet anyone? In the meantime, I'm starting with this Wikipedia page, which nicely categorizes these objects based on 1 how many sets and 2 how many binary operations: I notice at the beginning of the list is "set", defined as "degenerate algebraic structure having no operations". I have some questions about this:. From looking at the Wikipedia page, it seems that they are considering sets with no operations whatsoever.

This makes sense, since sets themselves don't come with a natural binary operation.

The question is therefore about set theory, and depends on the formal way that you approach sets-- that is, on the axioms of the set theory to which you subscribe. As far as I know disclaimer: I am no expert most widely accepted set theories allow for notions of union, intersection, cartesian product, and complement that is, these "operations" are axioms or can be constructed from axioms. This is the definition of a binary operation. The cartesian product is not defined using binary operations or functions, so these definitions are not circular.

Union, intersection, symmetric difference and relative complement are binary operations on any collection of sets closed under these operations. They are not generally defined on the elements of a single set.

Algebraic Structure: Are Set Operations Considered Binary Operations? - Mathematics Stack Exchange

The cartesian product and disjoint union are also binary operations defined on collections of sets. By posting your answer, you agree to the privacy policy and terms of service.

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Binary operation - Wikipedia

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Algebraic structures

Are Set Operations Considered Binary Operations? I have some questions about this: Are the well-known "set" operations "union" and "intersection" considered binary operations? Does the notion of "set" at the beginning of this list of algebraic structures, considered as having no binary operations include a set along with union and intersection? Is this a more complicated structure with a different name, or do we "always" have access to union and intersection when we are talking about sets?

Similarly as the previous questions, but what about the notions of "complement" and "set difference", i. And are they always included with the notion of a set or do they create an additional structure, e. Is this considered a "binary operation"? If so, then binary operations defined in terms of the cartesian product. However, isn't the cartesian product itself considered a binary operation? Lastly, let's remember that a function itself a function between two sets is defined as a special relation between those two sets, and a relation is a subset of the cartesian product of the two sets.

So again, function and hence binary operation are defined in terms of cartesian product. Yet it seems like cartesian product IS a binary relation Thank you for your help clearing this up! Mathemanic 1, 1 8 These are all very satisfying answers to all of my questions.

As you probably figured out, I was making the mistake whacka pointed out: I was most of the time thinking of collections of sets e. Thank you very much!

Do keep in mind the distintion between a single set and a collection of sets. Whacka, thank you for the clarification.

I was thinking of collection of sets for some reason Must be because I've been thinking about sigma algebras and topologies lately. Also, thank you for pointing out a helpful distinction: Note that a function itself is a binary relation between two sets, so this implies that every binary operation is a binary relation, correct?

Thanks, this is great stuff. Is there a book that covers all of this sort of material, from the beginning? Mathemanic We usually think about very different algebraic structures in very different ways - that's why the subject material is split into many different areas. Ultimately though I think the best course of action is to become acquainted with the important types of algebraic structures in isolation first, and then piece them together later after some maturity is developed, instead of initiating a campaign for some kind of 'holistic' understanding.

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