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Math Help - Basic definition of binary operations

  1. #1
    Junior Member beebe's Avatar
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    Basic definition of binary operations

    Sorry this is such a basic question, but I'm studying all on my lonesome.

    I'm struggling with the function/mapping definition of binary operations. I get the idea that if  a,b \in A and a \star b \in A, then \star is a binary operation. How does f: A \times A \rightarrow A mean the same thing? It seems like if A were a set of individual numbers (such as the set \mathbb{Z}), then f would change A to a set of ordered pairs made from elements of A. Also, is f the binary operator in the definition?
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Basic definition of binary operations

    Just a notation a*b:=f(a,b) .
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  3. #3
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    Re: Basic definition of binary operations

    A binary operator is a function of two arguments. Often they are denoted by symbols like +, \times, etc. rather than letters and are written using infix notation, e.g., x+y instead of +(x,y).

    Quote Originally Posted by beebe View Post
    I get the idea that if  a,b \in A and a \star b \in A, then \star is a binary operation.
    Before a mathematical object can be used, it has to be declared, i.e., it should be given a definition or at least a type: an integer, a real, a function, a set, etc. Saying that \star is a binary operator on A means that \star is a function from A\times A to A. Declaring the type of \star should come before forming statements like a\star b\in A.

    Quote Originally Posted by beebe View Post
    How does f: A \times A \rightarrow A mean the same thing? It seems like if A were a set of individual numbers (such as the set \mathbb{Z}), then f would change A to a set of ordered pairs made from elements of A.
    I am not sure if I see what you mean, but if f:A\times A\to A and x\in A, then f(x) can be considered as a partially applied function A\to A, i.e., a set of ordered pairs. This is used in programming sometimes. However, when f:A\times A\to A, the domain of f consists of pairs of elements of A, so, strictly speaking, f has to be applied to the whole pair at once.
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  4. #4
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    Re: Basic definition of binary operations

    in some cases (particularly when A is finite), the function is f is listed explicitly, with rows labelled "a" and columns labelled "b". this gives a matrix with a*b in the i,j-th postion. such a matrix is called a multiplication table, and may be familiar to you from grade-school.
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  5. #5
    Junior Member beebe's Avatar
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    Re: Basic definition of binary operations

    Thanks. I misunderstood how the function notation works.
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