# of different binary ops. on a set S

  • Thread starter Unassuming
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In summary, the conversation discusses the number of binary operations that can be defined on a set S with 1, 2, 3, or n elements. The answer is 1 element = 1, 2 elements = 2^4, 3 elements = 3^9, and n elements = n^(n^2). The participants also mention their lack of familiarity with combo classes and come to an agreement on the answer.
  • #1
Unassuming
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Bear with me,

Haow mainy difffferent biiinary operarations can be dephined on the set S when it has 1,2,3 or n elementz?

Sorry I typed so wierd, I don't want anybody finding this answer.

Can I confirm with you guys/gals in here that the answer is (hopefully)

1 elem = 1,
2 elem = 2^4
3 elem = 3^9
n elem = n^(n^2)

I have never had any combo class so forgive me if I am wrong with this.
 
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  • #2
Hi Unassuming! :smile:

hmm … never come across that before …

if the question means what I think it does, then I agree with you …

there are n values to be assigned to n² combinations, so it's n^n². :smile:

(I think … :rolleyes:)
 

FAQ: # of different binary ops. on a set S

How many different binary operations can be defined on a given set S?

The number of different binary operations that can be defined on a set S depends on the cardinality (size) of the set. If S has n elements, then there can be up to n^(n^2) different binary operations on S.

What is a binary operation on a set?

A binary operation on a set is a mathematical function that takes two elements from the set as inputs and produces a single element from the set as an output. The operation can be any mathematical operation such as addition, multiplication, or composition.

How do you determine if a given binary operation is valid on a set S?

A binary operation is valid on a set S if it satisfies closure, associativity, and identity properties. Closure means that the operation must produce an element from the set when given any two elements from the set. Associativity means that the order in which the operation is performed does not matter. Identity means that there exists an element in the set that when operated on with any other element in the set, produces the latter element.

Can there be more than one valid binary operation on a set S?

Yes, there can be more than one valid binary operation on a set S. In fact, as mentioned earlier, there can be up to n^(n^2) different binary operations on a set with n elements. This means that there can be an infinite number of valid binary operations on an infinite set.

What is the difference between a binary operation and a unary operation?

A binary operation takes two elements from a set as inputs and produces a single element from the same set as an output. A unary operation, on the other hand, takes only one element from a set as an input and produces a single element from the same set as an output. In other words, a unary operation is a function that operates on a single element, while a binary operation is a function that operates on two elements.

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