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**knightstar****Member**- Registered: 2014-03-12
- Posts: 14

Is this a proper interpretation of these four terms? Am I understanding them correctly?

**Operation** is a function which produces an output from one or more operands

**Function** is a relation that exists between an input and output

**Relation (i.e. predicate)** is the output value of an operation

**Operand (i.e. argument)** is the input value of an operation

OTHER RELATED DEFINITIONS:

Value is a known numerical amount

Quantity is a known or unknown numerical amount

I'd appreciate any input. Thank you.

*Last edited by knightstar (2014-03-31 04:35:06)*

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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,562

I think a relation is the input value and the output value, if not more data than two entities, I'm not sure, plus the way they are interconnected. Wolfram Alpha says something about two things...

**igloo** **myrtilles** **fourmis**

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,426

hi knightstar,

Your definition for an operation looks ok to me.

Relation.

Not quite.

http://simple.wikipedia.org/wiki/Relation_(mathematics)

and

http://www.mathsisfun.com/sets/function.html

Objects in one set may be 'related' to objects in another set.

eg. If x^2 + y^2 = 25, then certain x coordinates are related to certain y coordinates by the relationship.

x=3 is related to y = 4. But x = -3 is also related to 4; and x = 3 is also related to y = -4; and x = -3 is related to y = -4.

If you plot points obeying the relationship, you'll get a circle radius 5, centred on (0,0).

http://www.mathsisfun.com/data/grapher-equation.html

A function, y, in terms of x, is a relation that is 'well-defined'; each value of x in the domain has exactly one value of y in the codomain.

The circle above is not a function, because most values of x lead to two values of y.

y = x^2 is a well defined function. (note that you can get values of y with two values of x such as 4 = (2)^2 and 4 = (-2)^2

That's ok because we always know what y we will get when applying the function.

The inverse would not be a function because 4 is related to both 2 and -2.

On a calculator, the square root button is made into a function by only giving the positive square root.

Operand is ok.

Predicate is overused in mathematics and could lead to misunderstandings. I avoid using it completely.

Bob

ps. But look also at what I said in this post:

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**ShivamS****Member**- Registered: 2011-02-07
- Posts: 3,555

A function f is a relationship between two sets: a gender X called the domain and a set Y called the range or set of values which is satisfied by a rule which assigns to each element of X a unique element of Y and which assigns each element of Y to one or more elements of X. In other words, a function f is a set of ordered pairs (x, y) where x is an element of a set X, y is an element of a set Y and no two pairs in f have the same first element.

*Last edited by ShivamS (2014-03-31 08:06:41)*

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**knightstar****Member**- Registered: 2014-03-12
- Posts: 14

bob bundy wrote:

...Your definition for an operation looks ok to me.

Relation.

Not quite....

Perhaps this is an improvement:

**Operation** is a function of an output from one or more inputs

**Function** is a relation of an output to one or more inputs

**Relation** is a collection of one or more outputs and one or more inputs within a set

ShivamS wrote:

In other words, a function f is a set of ordered pairs (x, y) where x is an element of a set X, y is an element of a set Y and no two pairs in f have the same first element.

Correct me if I'm wrong, but ordered pairs do not necessarily have input and output, but rather are simply an arranged pair.

*Last edited by knightstar (2014-03-31 09:08:43)*

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**ShivamS****Member**- Registered: 2011-02-07
- Posts: 3,555

An ordered pair is just a set of numbers (or rather, "mathematical objects"). They don't have to be "input and output", but that does not make a difference in the definition of a function.

A function is just a subset of the cartesian product X x (multiplication symbol) Y.

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**knightstar****Member**- Registered: 2014-03-12
- Posts: 14

ShivamS wrote:

An ordered pair is just a set of numbers (or rather, "mathematical objects"). They don't have to be "input and output", but that does not make a difference in the definition of a function.

A function is just a subset of the cartesian product X x (multiplication symbol) Y.

I was under the impression that a function required an input *and* an output.

Though I was curious about(assuming that a function did in fact require an input, relation, and an output) how unary functions could be explained.

For example, -2,

input, **-** (?)

relation, **2** (??)

output, **-2** (???)

I got the idea that a function has an input, relation, and output from the link provided by Bob Bundy above.... and now directly below:

http://www.mathsisfun.com/sets/function.htm

*Last edited by knightstar (2014-03-31 09:10:32)*

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**ShivamS****Member**- Registered: 2011-02-07
- Posts: 3,555

"Input output" is a common analogy used but it's much better to think about functions using the more formal definition. There is no problem in thinking about it that way, but I like the more rigorous definition a lot more (it makes a lot more sense).

A function is a relation that uniquely associates members of one set with members of another set. Another way of thinking which means the same thing is that a function from A to B is an object f such that every a in A is uniquely associated with an object f(a) in B.

*Last edited by ShivamS (2014-03-31 08:50:27)*

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**knightstar****Member**- Registered: 2014-03-12
- Posts: 14

Before I go, I feel compelled to ask just one more question. Is there any difference what-so-ever between the operand and the input? Every time I've used the term thus far I've noticed that the term "input" would have

sufficed. It seems to make things a little more clear to use "input", especially because the antonym to input is output and the antonym to operand seems to be output as well. Any thoughts?

*Last edited by knightstar (2014-03-31 09:11:57)*

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**ShivamS****Member**- Registered: 2011-02-07
- Posts: 3,555

An operand is an object upon which a mathematical operation is performed. Sometimes you can use them interchangeably, but they have different meanings.

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**knightstar****Member**- Registered: 2014-03-12
- Posts: 14

ShivamS wrote:

An operand is an object upon which a mathematical operation is performed. Sometimes you can use them interchangeably, but they have different meanings.

An operand is an object upon which a mathematical operation is performed... which is not always an input? Can you give an example when an operand is not an input.

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**eigenguy****Member**- Registered: 2014-03-18
- Posts: 78

ShivamS - slow down a bit. Knightstar needs to learn how to walk before he can run.

knightstar - there a two types of terminology in mathematics:

1. Terms that refer to specific mathematical objects, such as "one", "two", "addition", etc. These must have very precise definitions.

2. Terms that are used to describe mathematical elements or processes, such as "operand", "addend", "input", "output", etc. These tend to be more loose in definition, like words in ordinary language. How they are defined may differ somewhat from one person to another.

How I would define your terms - aiming at your current mathematical level:

Function - a process that starts with an ordered collection of values (the "inputs") and produces a single value from them (the "output"). It is critical here that if the same collection of inputs is used again, then the same output will be produced. If the function takes only one input value, then it is called a "function of one variable". If it requires an ordered pair of input values, it is called a "function of two variables", etc.

Operation = Function. The only difference between the processes that we call Operations and those we call Functions is how we think of them. We think of operations as ways of combining objects (particularly numbers) to produce other objects. We think of Functions as "machines" spit out new objects when other objects are fed into them. In particular, Functions are thought of as being objects themselves. However, in truth, they are really the same thing: A unary operation is a function of one variable. A binary operation is a function of two variables, etc.

Operand = one of the inputs to an operation.

Relation = a logical function. That is, one whose output value is either "true" or "false". For example, "=" is a relation of two variables. "Is an integer" is a relation on only one variable.

value = the thing that a variable represents. This may or may not be numeric: in the expression x[sup]2[/sup] - 1, the values of x can be assumed to be numeric. But in the expression

, the values of A and B are apparently sets, not number.quantity = a real numeric value (not just a number, but a member of the Real numbers - no imaginary or non-real complex numbers allowed).

"Having thus refreshed ourselves in the oasis of a proof, we now turn again into the desert of definitions." - Bröcker & Jänich

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**ShivamS****Member**- Registered: 2011-02-07
- Posts: 3,555

I don't understand these education systems... What is wrong with teaching set theory first? (not aimed at eigenguy, I understand I should have given the OP an easier definition)

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**eigenguy****Member**- Registered: 2014-03-18
- Posts: 78

They tried that in 60's and early 70's. It was referred to as "the new math", and it was a dismal failure. Why? One reason was because most of the elementary school teachers never adequately understood what it was they were teaching, so they were unable to communicate it to their students. I learned the basic ideas of set theory in grade school (I was fortunate I guess in having some teachers who understood that much), but I was in high school before it became more than a rather boring and pointless thing we talked about a little each year. (I will admit that it helped in high school that I already knew the terminology when we actually started to make use of it.) And it wasn't until college that I came to know its true usefulness and power.

"Having thus refreshed ourselves in the oasis of a proof, we now turn again into the desert of definitions." - Bröcker & Jänich

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**ShivamS****Member**- Registered: 2011-02-07
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That's why I love principles of math (oakley/allendoerfer). Starts from set theory and builds everything from it.

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**eigenguy****Member**- Registered: 2014-03-18
- Posts: 78

I know. I feel the same way about Bartle's "Elements of Real Analysis", which is the book that caused me to start following Mathematics as a pursuit in and of itself. It doesn't build a model of the real numbers as like Oakley & Allendoefer apparently does (from comments you've made about it in other threads), but the teacher of that class took us through the whole process - from cardinal numbers to Dedekind cuts - before we even cracked the textbook, so I never felt the lack. I still feel that Bartle has the best, most instructional, set of exercises of any textbook I have ever read.

"Having thus refreshed ourselves in the oasis of a proof, we now turn again into the desert of definitions." - Bröcker & Jänich

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**ShivamS****Member**- Registered: 2011-02-07
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The Elements of Real Analysis is the text that I used in after I learned analysis through Rudin. It's an outstanding book, in my opinion. In terms of mathematical maturity, it is aimed about midway between Spivak's Calculus and Rudin's baby analysis book. It isn't as terse as Rudin's or as conversational as Spivak's. A good illustration of the relative level of sophistication is the treatment of topology in the book. Bartle does cover the standard elementary topology used in analysis: compactness, connectedness, Heine-Borel, equicontinuity, etc., all in the special case of R^n. He doesn't talk about general metric spaces at all, except in an exercise or two. Both Bartle and Rudin present the Riemann-Stieltjes integral, with the Riemann integral as a special case. Rudin has a rather perfunctory chapter at the end of the book covering the Lebesgue theory; Bartle doesn't. Bartle has a generous amount of material on Fourier series, perhaps about as much as you can do with Riemann integrals. Bartle has a lot of really good problem sets, both more generous in number and less difficult than Rudin's. Most of these are extensions or applications of the theory covered in the text. They are, I think, the "right" level of challenge for undergraduate analysis - substantial enough that you will come away with a good understanding of the material, but not so hard as to be unreasonable. I spent many hours each week on the problem sets, but in the end was usually able to do all of them. In many of the chapters, Bartle also has a set of "projects", which are a series of exercises developing a new topic. These are really great, a highlight of the book. For example, in chapter 30 ("Existence of the Integral"), he has four projects in addition to the regular problem set. The first project (in 7 parts) develops the logarithm function and its properties starting from the integral definition. The second project (in 6 parts) does the same thing with sine and cosine. The third project develops the Wallis product in 5 parts. The fourth does the same with the Stirling formula, in 6 parts. Overall, I think very highly of Bartle's Elements book. If you forced me to get rid of all of my undergraduate analysis books except one, I would keep baby Rudin because it is less cluttered with pedagogy and serves as a better reference - and for the same reason, I don't especially recommend Rudin as a good first book to learn from. If you let me keep one more, it would be Bartle (maybe Berberian too?).

*Last edited by ShivamS (2014-03-31 13:20:17)*

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**knightstar****Member**- Registered: 2014-03-12
- Posts: 14

Thanks for the reading suggestions. As for the definitions, I believe it's best for me to learn a little more and then come back to it.

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**ShivamS****Member**- Registered: 2011-02-07
- Posts: 3,555

Well, not really. It was my mistake you confuse you - at this point you shouldn't get bogged down by the more rigorous definitions - you will learn them later. You should be able to learn the more common definitions of a function and other terms right now.

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**eigenguy****Member**- Registered: 2014-03-18
- Posts: 78

These aren't reading suggestions for you, knightstar - at least not for a few more years. They are excellent books (certainly Bartle is, and I have no doubt that ShirvamS is entirely correct about the others), but you are not ready for any of them. They all expect you to have mastered a number of basic concepts and terminology of mathematics before you begin, and your questions here are about those basics.

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**ShivamS****Member**- Registered: 2011-02-07
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Of course Bartle is very advanced for him at this point, but I am pretty sure he can handle "Principles of Mathematics" by Oakley and Allendoerfer (the reason I keep mentioning the authors is being this is often confused with the book by Russel). Really, the only prerequisite to it is addition. However, he probably wouldn't be interested in the chapters on groups, limits, boolean algebra and calculus, but he would find the chapters on set theory, mathematical proof/logic, properties of numbers, equations and functions very useful.

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