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I'm reading the 4th paragraph on page 8, under section 1.2 Fields, in the following book https://drive.google.com/file/d/1KQ7dbLXI4x39VwZovTL0DKRsZwt_i3Vt/edit "linear algebra done openly".

the summary is:

a function $f$ is a relationship between sets, say $A$ and $B$...we denote this function relation as $f: A \rightarrow B$... $A\times B$ denotes the set of ordered pairs of elements from $A$ and $B$... An operation is a function of the form $f: A \times B \rightarrow C$. One should think of an operation as a process of bringing two objects together and creating a third operation.

what does:

"An operation is a function of the form $f: A \times B \rightarrow C$. One should think of an operation as a process of bringing two objects together and creating a third operation." exactly mean? what would a good example look like?

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2 Answers 2

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An "operation" takes two objects, and combines them to produce a third object. For example, addition is an operation on (for example) the real numbers; if I have two real numbers like $1$ and $2$, addition combines them to form $1 + 2 = 3$.

The quotation in bold is essentially saying that any operation can be written as a function whose domain is a cartesian product space. Using my example, one can represent addition as a function on the set of pairs of real numbers $$f_+ : \mathbb{R}\times\mathbb{R} \rightarrow \mathbb{R},$$ whose action on pairs of numbers is to add them: $$f_+(x,y) = x + y.$$ (Note: when your textbook says operation, they really mean "binary operation". In general an operation can take $n$ inputs)

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He’s providing a rather abstract way of looking at operations like addition and multiplication.

We can use addition of real numbers as an example. The operation of addition can be regarded as a function that receives two real numbers as input, and produces a real number as output. If we denote this function by $f$, then $f$ is a mapping from $\mathbb R \times \mathbb R$ to $\mathbb R$. It’s input is a pair of numbers $(a,b) \in \mathbb R \times \mathbb R$ and it’s output is $f(a,b) = a+b \in \mathbb R$.

Similarly, the dot product operation for $3D$ vectors is a function that maps $\mathbb R^3 \times \mathbb R^3$ to $\mathbb R$: given two vectors $a \in \mathbb R^3$ and $b\in \mathbb R^3$, the dot product function produces a number $a \cdot b \in \mathbb R$.

There are several more examples on the next few pages of your textbook.

This sort of abstraction doesn’t seem very helpful, to me, but it might be useful to you if you want to pass your linear algebra course.

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