Is it possible to multiply a set by a natural number? [on hold]
Say I have a set $S={1, 4, 10, 7}$. Could I then multiply $S$ by $3$? Would that then look like $3S={3, 12, 30, 21}$? Any help would be really appreciated.
discrete-mathematics elementary-set-theory
New contributor
put on hold as off-topic by BlueRaja - Danny Pflughoeft, DRF, Cesareo, amWhy, Adrian Keister yesterday
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – DRF, amWhy, Adrian Keister
If this question can be reworded to fit the rules in the help center, please edit the question.
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Say I have a set $S={1, 4, 10, 7}$. Could I then multiply $S$ by $3$? Would that then look like $3S={3, 12, 30, 21}$? Any help would be really appreciated.
discrete-mathematics elementary-set-theory
New contributor
put on hold as off-topic by BlueRaja - Danny Pflughoeft, DRF, Cesareo, amWhy, Adrian Keister yesterday
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – DRF, amWhy, Adrian Keister
If this question can be reworded to fit the rules in the help center, please edit the question.
11
Yes you could just define $$n{a_i}={na_i}$$
– clathratus
yesterday
13
You can certainly take a set of numbers $S= {1,4,10,7}$ and say "Hey, I'm going to multiply every element by $3$ and get the set ${3,12,20,21}$ and I'm going to call that set $3S$". And you can say "I'm going to refer to that as multiplying a set by a number, any objections? No? Good."
– fleablood
yesterday
5
I'm not sure what is being asked here. You can most certainly do anything you want in maths (assuming logically valid). Are you asking about usefulness of this construction?
– freakish
yesterday
21
@fleablood I do have an objection since $3 times 10 = 30$
– F.Carette
yesterday
1
Hello and welcome to Math Stack exchange. What is your background? Where did you come across this notation or problem? Especially for a problem like this context is very important.
– DRF
yesterday
|
show 5 more comments
Say I have a set $S={1, 4, 10, 7}$. Could I then multiply $S$ by $3$? Would that then look like $3S={3, 12, 30, 21}$? Any help would be really appreciated.
discrete-mathematics elementary-set-theory
New contributor
Say I have a set $S={1, 4, 10, 7}$. Could I then multiply $S$ by $3$? Would that then look like $3S={3, 12, 30, 21}$? Any help would be really appreciated.
discrete-mathematics elementary-set-theory
discrete-mathematics elementary-set-theory
New contributor
New contributor
edited yesterday
clathratus
3,347331
3,347331
New contributor
asked yesterday
Hunter KimuraHunter Kimura
522
522
New contributor
New contributor
put on hold as off-topic by BlueRaja - Danny Pflughoeft, DRF, Cesareo, amWhy, Adrian Keister yesterday
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – DRF, amWhy, Adrian Keister
If this question can be reworded to fit the rules in the help center, please edit the question.
put on hold as off-topic by BlueRaja - Danny Pflughoeft, DRF, Cesareo, amWhy, Adrian Keister yesterday
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – DRF, amWhy, Adrian Keister
If this question can be reworded to fit the rules in the help center, please edit the question.
11
Yes you could just define $$n{a_i}={na_i}$$
– clathratus
yesterday
13
You can certainly take a set of numbers $S= {1,4,10,7}$ and say "Hey, I'm going to multiply every element by $3$ and get the set ${3,12,20,21}$ and I'm going to call that set $3S$". And you can say "I'm going to refer to that as multiplying a set by a number, any objections? No? Good."
– fleablood
yesterday
5
I'm not sure what is being asked here. You can most certainly do anything you want in maths (assuming logically valid). Are you asking about usefulness of this construction?
– freakish
yesterday
21
@fleablood I do have an objection since $3 times 10 = 30$
– F.Carette
yesterday
1
Hello and welcome to Math Stack exchange. What is your background? Where did you come across this notation or problem? Especially for a problem like this context is very important.
– DRF
yesterday
|
show 5 more comments
11
Yes you could just define $$n{a_i}={na_i}$$
– clathratus
yesterday
13
You can certainly take a set of numbers $S= {1,4,10,7}$ and say "Hey, I'm going to multiply every element by $3$ and get the set ${3,12,20,21}$ and I'm going to call that set $3S$". And you can say "I'm going to refer to that as multiplying a set by a number, any objections? No? Good."
– fleablood
yesterday
5
I'm not sure what is being asked here. You can most certainly do anything you want in maths (assuming logically valid). Are you asking about usefulness of this construction?
– freakish
yesterday
21
@fleablood I do have an objection since $3 times 10 = 30$
– F.Carette
yesterday
1
Hello and welcome to Math Stack exchange. What is your background? Where did you come across this notation or problem? Especially for a problem like this context is very important.
– DRF
yesterday
11
11
Yes you could just define $$n{a_i}={na_i}$$
– clathratus
yesterday
Yes you could just define $$n{a_i}={na_i}$$
– clathratus
yesterday
13
13
You can certainly take a set of numbers $S= {1,4,10,7}$ and say "Hey, I'm going to multiply every element by $3$ and get the set ${3,12,20,21}$ and I'm going to call that set $3S$". And you can say "I'm going to refer to that as multiplying a set by a number, any objections? No? Good."
– fleablood
yesterday
You can certainly take a set of numbers $S= {1,4,10,7}$ and say "Hey, I'm going to multiply every element by $3$ and get the set ${3,12,20,21}$ and I'm going to call that set $3S$". And you can say "I'm going to refer to that as multiplying a set by a number, any objections? No? Good."
– fleablood
yesterday
5
5
I'm not sure what is being asked here. You can most certainly do anything you want in maths (assuming logically valid). Are you asking about usefulness of this construction?
– freakish
yesterday
I'm not sure what is being asked here. You can most certainly do anything you want in maths (assuming logically valid). Are you asking about usefulness of this construction?
– freakish
yesterday
21
21
@fleablood I do have an objection since $3 times 10 = 30$
– F.Carette
yesterday
@fleablood I do have an objection since $3 times 10 = 30$
– F.Carette
yesterday
1
1
Hello and welcome to Math Stack exchange. What is your background? Where did you come across this notation or problem? Especially for a problem like this context is very important.
– DRF
yesterday
Hello and welcome to Math Stack exchange. What is your background? Where did you come across this notation or problem? Especially for a problem like this context is very important.
– DRF
yesterday
|
show 5 more comments
3 Answers
3
active
oldest
votes
Sure, we sometimes for example denote the set of even integers by $2Bbb Z={dots,-4,-2,0,2,4,dots}$, while the set of integers is $Bbb Z={dots,-2,-1,0,1,2,dots}$.
add a comment |
Yes..you have already defined the operation..a scalar multiplication on a set.
add a comment |
In fact, this is closely related to images of functions:
Let $f: A to B$ be a function between two sets and $A' subseteq A$, then we define
$$f[A'] := {f(a) | a in A'}quad left(= {b in B | exists a' in A'. f(a') = b}right)$$
Your idea is a specific instance:
Let $A' = {1, 4, 10, 7} subseteq mathbb{N} =: A$ and $f: mathbb{N} to mathbb{N}$ be the operation "multiply by three", then
$$f[A'] = {3, 12, 30, 21}$$
The other answer hints at the notation $NA'$, e.g. $3A'$.
Note that in general only $|A'| geq |f[A']|$ holds, e.g. consider $0mathbb{Z} = {0}$. However, with this "function framework", it is easy to state a criterion for when the equality holds:
If $f$ is injective on $A'$, then $|A'| = |f[A']|$.
Or in words: "If you map different elements in $A'$ to different elements in $B$, then certainly, we cannot lose elements due to duplicates in the set $f[A']$."
Can you see why $|2mathbb{Z}| = |mathbb{Z}|$? This proves "there are as many even numbers as integers". (Of course, you have to first define the cardinality function $|cdot|$ for arbitrary infinite sets first.)
add a comment |
3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
Sure, we sometimes for example denote the set of even integers by $2Bbb Z={dots,-4,-2,0,2,4,dots}$, while the set of integers is $Bbb Z={dots,-2,-1,0,1,2,dots}$.
add a comment |
Sure, we sometimes for example denote the set of even integers by $2Bbb Z={dots,-4,-2,0,2,4,dots}$, while the set of integers is $Bbb Z={dots,-2,-1,0,1,2,dots}$.
add a comment |
Sure, we sometimes for example denote the set of even integers by $2Bbb Z={dots,-4,-2,0,2,4,dots}$, while the set of integers is $Bbb Z={dots,-2,-1,0,1,2,dots}$.
Sure, we sometimes for example denote the set of even integers by $2Bbb Z={dots,-4,-2,0,2,4,dots}$, while the set of integers is $Bbb Z={dots,-2,-1,0,1,2,dots}$.
answered yesterday
John DoeJohn Doe
10.9k11238
10.9k11238
add a comment |
add a comment |
Yes..you have already defined the operation..a scalar multiplication on a set.
add a comment |
Yes..you have already defined the operation..a scalar multiplication on a set.
add a comment |
Yes..you have already defined the operation..a scalar multiplication on a set.
Yes..you have already defined the operation..a scalar multiplication on a set.
answered yesterday
ershersh
24310
24310
add a comment |
add a comment |
In fact, this is closely related to images of functions:
Let $f: A to B$ be a function between two sets and $A' subseteq A$, then we define
$$f[A'] := {f(a) | a in A'}quad left(= {b in B | exists a' in A'. f(a') = b}right)$$
Your idea is a specific instance:
Let $A' = {1, 4, 10, 7} subseteq mathbb{N} =: A$ and $f: mathbb{N} to mathbb{N}$ be the operation "multiply by three", then
$$f[A'] = {3, 12, 30, 21}$$
The other answer hints at the notation $NA'$, e.g. $3A'$.
Note that in general only $|A'| geq |f[A']|$ holds, e.g. consider $0mathbb{Z} = {0}$. However, with this "function framework", it is easy to state a criterion for when the equality holds:
If $f$ is injective on $A'$, then $|A'| = |f[A']|$.
Or in words: "If you map different elements in $A'$ to different elements in $B$, then certainly, we cannot lose elements due to duplicates in the set $f[A']$."
Can you see why $|2mathbb{Z}| = |mathbb{Z}|$? This proves "there are as many even numbers as integers". (Of course, you have to first define the cardinality function $|cdot|$ for arbitrary infinite sets first.)
add a comment |
In fact, this is closely related to images of functions:
Let $f: A to B$ be a function between two sets and $A' subseteq A$, then we define
$$f[A'] := {f(a) | a in A'}quad left(= {b in B | exists a' in A'. f(a') = b}right)$$
Your idea is a specific instance:
Let $A' = {1, 4, 10, 7} subseteq mathbb{N} =: A$ and $f: mathbb{N} to mathbb{N}$ be the operation "multiply by three", then
$$f[A'] = {3, 12, 30, 21}$$
The other answer hints at the notation $NA'$, e.g. $3A'$.
Note that in general only $|A'| geq |f[A']|$ holds, e.g. consider $0mathbb{Z} = {0}$. However, with this "function framework", it is easy to state a criterion for when the equality holds:
If $f$ is injective on $A'$, then $|A'| = |f[A']|$.
Or in words: "If you map different elements in $A'$ to different elements in $B$, then certainly, we cannot lose elements due to duplicates in the set $f[A']$."
Can you see why $|2mathbb{Z}| = |mathbb{Z}|$? This proves "there are as many even numbers as integers". (Of course, you have to first define the cardinality function $|cdot|$ for arbitrary infinite sets first.)
add a comment |
In fact, this is closely related to images of functions:
Let $f: A to B$ be a function between two sets and $A' subseteq A$, then we define
$$f[A'] := {f(a) | a in A'}quad left(= {b in B | exists a' in A'. f(a') = b}right)$$
Your idea is a specific instance:
Let $A' = {1, 4, 10, 7} subseteq mathbb{N} =: A$ and $f: mathbb{N} to mathbb{N}$ be the operation "multiply by three", then
$$f[A'] = {3, 12, 30, 21}$$
The other answer hints at the notation $NA'$, e.g. $3A'$.
Note that in general only $|A'| geq |f[A']|$ holds, e.g. consider $0mathbb{Z} = {0}$. However, with this "function framework", it is easy to state a criterion for when the equality holds:
If $f$ is injective on $A'$, then $|A'| = |f[A']|$.
Or in words: "If you map different elements in $A'$ to different elements in $B$, then certainly, we cannot lose elements due to duplicates in the set $f[A']$."
Can you see why $|2mathbb{Z}| = |mathbb{Z}|$? This proves "there are as many even numbers as integers". (Of course, you have to first define the cardinality function $|cdot|$ for arbitrary infinite sets first.)
In fact, this is closely related to images of functions:
Let $f: A to B$ be a function between two sets and $A' subseteq A$, then we define
$$f[A'] := {f(a) | a in A'}quad left(= {b in B | exists a' in A'. f(a') = b}right)$$
Your idea is a specific instance:
Let $A' = {1, 4, 10, 7} subseteq mathbb{N} =: A$ and $f: mathbb{N} to mathbb{N}$ be the operation "multiply by three", then
$$f[A'] = {3, 12, 30, 21}$$
The other answer hints at the notation $NA'$, e.g. $3A'$.
Note that in general only $|A'| geq |f[A']|$ holds, e.g. consider $0mathbb{Z} = {0}$. However, with this "function framework", it is easy to state a criterion for when the equality holds:
If $f$ is injective on $A'$, then $|A'| = |f[A']|$.
Or in words: "If you map different elements in $A'$ to different elements in $B$, then certainly, we cannot lose elements due to duplicates in the set $f[A']$."
Can you see why $|2mathbb{Z}| = |mathbb{Z}|$? This proves "there are as many even numbers as integers". (Of course, you have to first define the cardinality function $|cdot|$ for arbitrary infinite sets first.)
answered yesterday
ComFreekComFreek
5321411
5321411
add a comment |
add a comment |
11
Yes you could just define $$n{a_i}={na_i}$$
– clathratus
yesterday
13
You can certainly take a set of numbers $S= {1,4,10,7}$ and say "Hey, I'm going to multiply every element by $3$ and get the set ${3,12,20,21}$ and I'm going to call that set $3S$". And you can say "I'm going to refer to that as multiplying a set by a number, any objections? No? Good."
– fleablood
yesterday
5
I'm not sure what is being asked here. You can most certainly do anything you want in maths (assuming logically valid). Are you asking about usefulness of this construction?
– freakish
yesterday
21
@fleablood I do have an objection since $3 times 10 = 30$
– F.Carette
yesterday
1
Hello and welcome to Math Stack exchange. What is your background? Where did you come across this notation or problem? Especially for a problem like this context is very important.
– DRF
yesterday