A function f is a relationship between sets, say A and B...we denote this function...

Roger Smith

Roger Smith

Answered

2022-01-14

A function f is a relationship between sets, say A and B...we denote this function relation as f:AB
A×B denotes the set of ordered pairs of elements from A and B... An operation is a function of the form f:A×BC. 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×BC. 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?

Answer & Explanation

Toni Scott

Toni Scott

Expert

2022-01-15Added 32 answers

Step 1
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.
Step 2
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+:R×RR,
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)
Ana Robertson

Ana Robertson

Expert

2022-01-16Added 26 answers

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 R×R to R. It’s input is a pair of numbers (a,b)R×R and it’s output is f(a,b)=a+bR.
Similarly, the dot product operation for 3D vectors is a function that maps R3×R3 to R : given two vectors aR3 and bR3, the dot product function produces a number abR.
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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