What does [math] f: [/ math] mean in math?

Updated on : December 4, 2021 by Kai Young



What does [math] f: [/ math] mean in math?

The symbols 'math f: / math' can be read as "math f / math is a mapping function." If it reads as a sparse sentence, that's correct: normally, you'd see those symbols in the context of something like math f: A \ a B / math, which can be read as "math f / math is a function that maps from math A / math to math B / math "(or, shorter," math f / math maps math A / math a math B / math ”), and you might see something else like math x \ mapsto f (x) / math (where math f (x) / math could be like such,

It's very common for you to see something like math f: \ mathbb {R} \ to \ mathbb {R} / math, math f (x) = e ^ x / math (or something like that! also write that math x \ map / math

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The symbols 'math f: / math' can be read as "math f / math is a mapping function." If it reads as a sparse sentence, that's correct: normally, you'd see those symbols in the context of something like math f: A \ a B / math, which can be read as "math f / math is a function that maps from math A / math to math B / math "(or, shorter," math f / math maps math A / math a math B / math ”), and you might see something else like math x \ mapsto f (x) / math (where math f (x) / math could be like such,

It's very common for you to see something like math f: \ mathbb {R} \ to \ mathbb {R} / math, math f (x) = e ^ x / math (or something like that! also writes that math x \ mapsto e ^ x / math for this function), which says that math f / math is a function that takes members of math \ mathbb {R} / math (so-called "real numbers"), and returns members of math \ mathbb {R} / math, according to the rule that if you give math x / math, the output is math e ^ x / math.

Thanks for the A2A!

Now in normal math, you've probably learned that a function is something like:

math f (x) = x ^ 2 / math

Now this is a function, but a field in mathematics called Set Theory is trying to define everything in mathematics. They were able to define functions in terms of sets.

Let's say you have two sets, set A and set B. A set is just a collection of objects. Now let's say you have some function that takes everything in set A, does some magic on those inputs, and spits out elements of set B. You could write the function like this:

math f \ ,: \, A \ a B / math or math f_ {A \ a B} / math

(I started learning set theory around 5 days

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Thanks for the A2A!

Now in normal math, you've probably learned that a function is something like:

math f (x) = x ^ 2 / math

Now this is a function, but a field in mathematics called Set Theory is trying to define everything in mathematics. They were able to define functions in terms of sets.

Let's say you have two sets, set A and set B. A set is just a collection of objects. Now let's say you have some function that takes everything in set A, does some magic on those inputs, and spits out elements of set B. You could write the function like this:

math f \ ,: \, A \ a B / math or math f_ {A \ a B} / math

(I started learning set theory about 5 days ago, so please correct me if I'm wrong)

math f / math means "function".

A function in mathematics is a relation that assigns a value mathematics x / mathematics in a domain to another value mathematics and / mathematics in a range, and only that value.

More information: Function (mathematics) - Wikipedia

The notation math f: / math represents a function that is defined in the scope of a domain and range. You would see that it is used like this: math f: x \ rightarrow y / math.

It is used to mean "function", which generally means a specific mathematical action that must be performed on a variable to give a specific result. See the link below for a better explanation Function (math) - Wikipedia. This is something that you will find introduced into pre-calculus courses and it becomes increasingly complex as you progress through math. Make sure you have the basics firmly in your mind, or you will quickly get lost.

It means "function".

For example, f (c) = mc ^ 2 means "the function of c is mxcx c".

Addendum: All other answers are also correct. I just thought that if you are asking such a simple question, you will want a simple answer.

Function (math) - Wikipedia

math f: A \ right arrow B / math Denotes a mapping from set A to set B.

This basically associates each element of set B to one or more elements of set A using a well-defined relation / expression. If it's the other way around, we call it a "Relation" denoted by math R: A \ rightarrow B / math

It's just a letter. You can make it mean whatever you want. What happens is that people have to use it to represent "function".

It could be a boundary operator on a string complex.

For example, if we have math \ partial_n: C_n \ right arrow C_ {n-1} / math, then the nth homology group can be defined as math H_n = \ frac {Z_n} {B_n } / math where math Z_n = \ text {Ker} \ partial_n / math are the cycles and math B_n = \ text {Im} \ partial_ {n + 1} / math are Limits. Homology groups informally tell which n-cycles are not n-limits. In a torus, you can find combinations of cycles (consisting of lines) that are not the boundary of a suitable region on the torus's surface (one without holes and so on). The surface of a torus is two-dimensional, thus a boundary of a region on the surface of the torus.

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It could be a boundary operator on a string complex.

For example, if we have math \ partial_n: C_n \ right arrow C_ {n-1} / math, then the nth homology group can be defined as math H_n = \ frac {Z_n} {B_n } / math where math Z_n = \ text {Ker} \ partial_n / math are the cycles and math B_n = \ text {Im} \ partial_ {n + 1} / math are Limits. Homology groups informally tell which n-cycles are not n-limits. In a torus, you can find combinations of cycles (consisting of lines) that are not the boundary of a suitable region on the torus's surface (one without holes and so on). The surface of a torus is two-dimensional, so the boundary of a region on the surface of the torus is one-dimensional. Therefore, we are looking for closed curves in a torus that are not boundaries of closed regions on the torus' surface. Since the torus is the product of two circles, it is not very difficult to visually see what these cycles look like.

Note that this is purely informal and that the actual homology group calculations are adequate and rigorous. Another way of thinking about homology groups is that they describe gaps of a certain dimension. There are quite nice connections between differential form integration and cohomology groups (note the prefix "co").

So yeah, if the math \ partial / math is a limit operator, expect some good stuff. Also, if you use math \ partial / math or math d / math as the symbol for your limit operator and let it be the outer derivative, then you get to the connection between this symbol and "differentials" ( Differential forms).

Before giving an answer, I suggest two readings of Isaac Asimov: the stories "Profession" and "The feeling of power." Both stories are very, very related to this question.

You can read a short summary in Profession (novella) - Wikipedia and The Feeling of Power - Wikipedia. You can even read Isaac Asimov's comprehensive "Profession" in Profession.

Those stories present Asimov's vision of what the world would be like without learning math ... See for yourself, you'll eventually agree with Asimov.

A compelling reason to learn math, or at least to teach math in school, in all grades

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Before giving an answer, I suggest two readings of Isaac Asimov: the stories "Profession" and "The feeling of power." Both stories are very, very related to this question.

You can read a short summary in Profession (novella) - Wikipedia and The Feeling of Power - Wikipedia. You can even read Isaac Asimov's comprehensive "Profession" in Profession.

Those stories present Asimov's vision of what the world would be like without learning math ... See for yourself, you'll eventually agree with Asimov.

A compelling reason to learn mathematics, or at least to teach mathematics in school, at all grades and levels, is to develop the brain. Learning mathematics and thinking about mathematics provide certain abstraction skills that other academic disciplines do not offer. In the end, the math rearranges your neural links ... usually in a logical way.

Another compelling reason to teach math is as a way to select people who like math (that is, people who feel that math works not as a burden, but as a pleasure, yes there are nerds like this. ) and to select people who are very good at math. These two groups of people do not completely overlap, but there are many who like mathematics, and are also very good at mathematics, and these, when they discover this "skill", will probably pursue a career in hard science to take advantage of your gift".

This argument about the teaching of mathematics is valid for biology, chemistry, physics, athletics, history, French, German, Spanish, geography, philosophy, etc. a career that is in sync with your skills. I mean, and using a metaphor, I don't know how Usain Bolt performed in math, but he's a great runner; if he had never tried athletics, the world would have lost one of the greatest athletes of all time. The same reasoning can be applied to the great mathematicians, artists, politicians (are there any? :-)), scientists, athletes, etc… At some point, in their first years of life, they “tried” their work area of ​​choice and they became addicted.

Therefore, in the early stages of education we never know exactly what each child will need to know throughout his life. Therefore, we must provide an "average" education, which includes fragments from many areas of knowledge. In later stages of education, people specialize according to their natural or acquired abilities. This is how our educational system is organized, and in my humble opinion it is well organized. And mathematics, even middle and advanced mathematics, must be there, because it is one of the pillars, perhaps the fundamental one, of modern science and technology; and science and technology is the main factor behind the improvement of the quality of life of people in recent centuries ... Maybe in the end it kills us all, but that's another story :-)

There are two interpretations of the meaning of this symbol:

(1) Geometrically: Given the statement math -5 \ le -3 / math that we really mean, the integer -5 is to the left of the integer -3 on the number line. If we take math x \ le -3 / math we mean that the parameter math x / math includes -3 and can take any value to the left of -3 on the number line. In this context, the symbol has more to do with direction and measurement than size.

(2) Numerically: the symbol certainly has an underlying size context in this case. What we mean by math x \ le-3 / math in this sense is that the parameter math x / math cannot be greater than math -3 / math. This certainly has value when d

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There are two interpretations of the meaning of this symbol:

(1) Geometrically: Given the statement math -5 \ le -3 / math that we really mean, the integer -5 is to the left of the integer -3 on the number line. If we take math x \ le -3 / math we mean that the parameter math x / math includes -3 and can take any value to the left of -3 on the number line. In this context, the symbol has more to do with direction and measurement than size.

(2) Numerically: the symbol certainly has an underlying size context in this case. What we mean by math x \ le-3 / math in this sense is that the parameter math x / math cannot be greater than math -3 / math. This certainly has value when talking about quantities with physical significance. For example, the amount of money you budget for Christmas is no more than $ 1000, or in other words, math C \ le $ 1000 / math would be a budget line in my Excel or other account software coding .

It is when these meanings are combined that true mathematical power arises. The math statement | xy | \ le 5 / math tells us that the parameters math x / math and math y / math (perhaps measuring distances from the origin) are not greater than 5 units apart. That is, regardless of the distances that math x / math and math y / math represent, their difference is strictly within 5 units, always.

Hope it sheds some light, albeit minimal, on the use of the math \ le / math symbol.

What does "math! / Math" mean as in math n! / Math in math?

math n! / math, pronounced “n factorial” or “n bang” by some, is the result of multiplying all positive integers less than or equal to math n / math. Using another mathematical notation we have:

math n! = n \ times (n-1) \ times \ dotsm \ times3 \ times2 \ times1 / math

math \ quad = \ displaystyle \ prod_ {k = 1} ^ nk / math

math \ quad = \ displaystyle \ int_0 ^ {\ infty} x ^ ne ^ {- x} \, \ text {d} x \ equiv \ Gamma (n + 1) / math 1

math \ quad \ approx \ sqrt {2 \ pi n} (n / e) ^ n / math 2

By convention math 0! = 1 / math which allows its use in many places as the binomial coefficient:

math \ quad \ binom {n} {k} \ equiv \, ^ n \ text {C} _k = \ frac {n!} {k! (nk)!} / math

which is also the number of ways to choose math k / math different objects

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Footnotes

1 Gamma function 2 Stirling approximation

What does "math! / Math" mean as in math n! / Math in math?

math n! / math, pronounced “n factorial” or “n bang” by some, is the result of multiplying all positive integers less than or equal to math n / math. Using another mathematical notation we have:

math n! = n \ times (n-1) \ times \ dotsm \ times3 \ times2 \ times1 / math

math \ quad = \ displaystyle \ prod_ {k = 1} ^ nk / math

math \ quad = \ displaystyle \ int_0 ^ {\ infty} x ^ ne ^ {- x} \, \ text {d} x \ equiv \ Gamma (n + 1) / math 1

math \ quad \ approx \ sqrt {2 \ pi n} (n / e) ^ n / math 2

By convention math 0! = 1 / math which allows its use in many places as the binomial coefficient:

math \ quad \ binom {n} {k} \ equiv \, ^ n \ text {C} _k = \ frac {n!} {k! (nk)!} / math

which is also the number of ways to choose math k / math different objects from a total of math n / math different objects.

Footnotes

1 Gamma function 2 Stirling approximation

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