Algеbraik opеratsiya tushunchasi va uning xossalari: kommutativlik, assotsiativlik, distributivlik va qisqaruvchanlik. Nеytral, yutuvchi va simmеtrik elеmеntlar

Definition of Binary Operation on a Set. A binary operation
on a non-empty set S is a mapping ∗ : S × S → S.
No more, no less! We usually write s ∗ s^r in place of the more formal
∗(s, s^r).
We have a wealth of examples available; we’ll review just a few of them here.

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  The familiar operations + and · are binary operations on our fa- vorite number systems: Z, Q, R, C.
  
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  Note that if S is the set of irrational numbers then neither + nor
  
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    defines a binary operation on S. (Why not?)
    
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  Note that subtraction − defines a binary operation on R.
  
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  Let Mat_n(R) denote the n×n matrices with real coefficients. Then both + (matrix addition) and · (matrix multiplication) define bi-nary operations on Mat_n(R).
  
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  Let S be any set and let F(S) = {functions : S → S}. Then function composition ◦ defines a binary operation on F(S). (This is a particularly important example.)
  
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  Let Vect₃(R) denote the vectors in 3-space. Then the vector cross product × is a binary operation on Vect₃(R). Note that the scalar product · does not define a binary operation on Vect₃(R).
  
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  Let A be a set and let 2^A be its power set. The operations ∩, ∪, and + (symmetric difference) are all important binary operations on 2^A.
  
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  Let S be a set and let Sym(S) be the set of all permutations on
  

S. Then function composition ◦ defines a binary operation on Sym(S). We really should prove this. Thus let σ, τ : S → S
be permutations; thus they are one-to-one and onto. We need to show that σ ◦ τ : S → S is also one-to-one and onto.
σ ◦ τ is one-to-one: Assume that s, s^r ∈ S and that σ ◦ τ (s) =
σ ◦ τ (s^r). Since σ is one-to-one, we conclude that τ (s) = τ (s^r).