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help-10
f4b6863439da4dd896e5804fe0ce0aab
565,602,777,275,826,200
Matas
01/18/2022 18:56:37
Hi, what is a surjective function?
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Matas
01/18/2022 18:57:00
I mean I've read the definition but still don't really understand it
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Matas
01/18/2022 18:57:20
https://cdn.discordapp.c…7838/unknown.png
unknown.png
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Deus_Vult
01/18/2022 19:04:44
well , given a certain domain and an range
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Deus_Vult
01/18/2022 19:05:09
the function is surjective if for ever element in its range there is such x from its domain that f(x)=y
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Deus_Vult
01/18/2022 19:05:32
take for example y=x^2 so that f : R -> R
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Deus_Vult
01/18/2022 19:06:11
the function isn't surjective, because there is not a real number x so that if you square it you get a negative number right?
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Deus_Vult
01/18/2022 19:07:23
so for example if you take -1 , an element of our range, (R) you can't find real number x such that x^2=-1
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Matas
01/18/2022 19:07:57
ok so the domain must go to all the values of the co-domain?
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Deus_Vult
01/18/2022 19:08:02
on the other hand, if you redefine the function so taht it maps f:R->[0,+infinity) , it is surjective now
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Deus_Vult
01/18/2022 19:08:24
yes exactly
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Deus_Vult
01/18/2022 19:08:56
maybe it will be more intuitive to think about it the other way around each value in the co domain has a value in its domain that is mapped to it.
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Matas
01/18/2022 19:09:09
ok
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0
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Deus_Vult
01/18/2022 19:09:26
made a small correction on the codomain definition
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Matas
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Matas
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Matas
01/18/2022 19:09:34
so the domain can have values not mapping to anything as long as some of the values map to all of the values of the co-domain
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Deus_Vult
01/18/2022 19:10:33
yes
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Deus_Vult
01/18/2022 19:10:36
that is correct
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Deus_Vult
01/18/2022 19:11:03
however do notice that it can only be surjective if the group of the co domain is smaller in its cardinality then the domain
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Deus_Vult
01/18/2022 19:11:57
like, if you have 4 values in your domain and 4 values in your co do main, then each value in your domain HAS to be mapped to a corresponding value in the co domain for it to be surjective
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Matas
01/18/2022 19:12:35
OK I understand, but why do I not understand the first example you gave
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Deus_Vult
01/18/2022 19:12:44
if the co domain is smaller, say 3, then you can map 3 valued from the domain to the 3 values of the codomain , thus achieving surjectivity
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Deus_Vult
01/18/2022 19:12:52
what exactly?
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Matas
01/18/2022 19:13:11
what does this mean
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Matas
01/18/2022 19:13:32
the r -> r part
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Matas
01/18/2022 19:13:41
set R to set R
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565,602,777,275,826,200
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3225
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565,602,777,275,826,200
Matas
01/18/2022 19:13:47
I mean its the function notation
1
0
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Deus_Vult
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Deus_Vult
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142
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244,129,917,732,651,000
Deus_Vult
01/18/2022 19:14:32
it just means that function f maps values from domain R to co domain R
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Matas
01/18/2022 19:14:43
okay
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Matas
01/18/2022 19:14:50
and R is not a set
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Matas
01/18/2022 19:14:52
right
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Deus_Vult
01/18/2022 19:15:06
it is a set
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Matas
01/18/2022 19:15:09
because then it would map set R to set R
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244,129,917,732,651,000
Deus_Vult
01/18/2022 19:15:10
it's the set of all Real numbers
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Deus_Vult
01/18/2022 19:15:17
yea it would
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Matas
01/18/2022 19:15:19
oh ok
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Deus_Vult
01/18/2022 19:16:12
you can map a set to itself after all
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Deus_Vult
01/18/2022 19:16:22
you can also map a function this way :
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Deus_Vult
01/18/2022 19:16:30
f: N -> R
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Deus_Vult
01/18/2022 19:17:45
aka a sequence
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Matas
01/18/2022 19:18:01
huh
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Matas
01/18/2022 19:18:07
I'm lost
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0
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244,129,917,732,651,000
Deus_Vult
01/18/2022 19:23:23
N->R means mapping natural numbers to real numbers
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Deus_Vult
01/18/2022 19:23:51
a seuqnece a1,a2,a3...an is a function that takes a natural number, being the index and maps it to some real number
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Deus_Vult
01/18/2022 19:24:17
a1=3.14 is essentially the mapping 1->pi
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Matas
01/18/2022 19:24:24
ok
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0
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Matas
01/18/2022 19:25:29
and bijective is when each value in the domain maps to a unique value in the co-domain
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Matas
01/18/2022 19:25:37
?
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Deus_Vult
01/18/2022 19:27:24
what you described is injectivity
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Deus_Vult
01/18/2022 19:27:40
a bijective function is one that is both Injective and surjective
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Matas
01/18/2022 19:28:20
ok
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Matas
01/18/2022 19:28:32
so both domain and co domain have the same cardinality?
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565,602,777,275,826,200
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false
142
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Matas
01/18/2022 19:29:32
since for it to be injective, it has to for each input have a unique output
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Matas
01/18/2022 19:29:56
and if the co-domain was larger, it would not be surjective
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Matas
01/18/2022 19:30:03
?
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Deus_Vult
01/18/2022 19:30:39
well yes but not exactly a more precise definition would be that if for values a and b f(a)=f(b) then a=b
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Deus_Vult
01/18/2022 19:30:57
the codomain must be of equal size or smaller then the domain
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Matas
01/18/2022 19:31:16
how
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Deus_Vult
01/18/2022 19:31:21
you can map a domain of 4 values to a co domain of 3 values so that the mapping is bijective
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565,602,777,275,826,200
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false
142
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565,602,777,275,826,200
Matas
01/18/2022 19:31:28
won't that mean that its not surjective anymore
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Deus_Vult
01/18/2022 19:31:50
say your domain is 1 2 3 4 and co domain is 1 2 3
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Matas
01/18/2022 19:31:50
oh wait no
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Deus_Vult
01/18/2022 19:32:02
I can map 1 -> 1 2->2 3->3
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Matas
01/18/2022 19:32:06
ok so 1 element maps to 2 things
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Deus_Vult
01/18/2022 19:32:20
the function I defined is both injective and surjective, and thus bijective
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Deus_Vult
01/18/2022 19:32:50
having an element map to 2 things defied the definition of a function
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Matas
01/18/2022 19:32:52
so not all values of the domain have to be used?
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Deus_Vult
01/18/2022 19:32:58
Nope
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Deus_Vult
01/18/2022 19:32:59
I mean
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Deus_Vult
01/18/2022 19:33:03
take for example 1/x
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Deus_Vult
01/18/2022 19:33:13
the value 0 in the domain is never used
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Deus_Vult
01/18/2022 19:33:29
such a point would be an incontinuity point
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Deus_Vult
01/18/2022 19:33:33
for example
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730,028,793,875,398,700
Blaxapate
01/18/2022 19:34:25
Wait, no; the domain of a function is exactly where it is defined
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Blaxapate
01/18/2022 19:34:53
A bijection between sets indeed imposes that the sets are of same cardinality
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730,028,793,875,398,700
Blaxapate
01/18/2022 19:35:49
Partial functions exist and are useful, but when talking about injection, surjection or bijection, this is for non-partial functions.
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730,028,793,875,398,700
Blaxapate
01/18/2022 19:36:23
The domain of 1/x is not R, it R* (R without 0)
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Matas
01/18/2022 19:37:01
ok right, so let me get this from the start, injective means that all the values in the domain map to a UNIQUE element in the co-domain
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Blaxapate
01/18/2022 19:37:34
yes
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Matas
01/18/2022 19:37:53
and surjective means that all the values of the co-domain have a mapping
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Deus_Vult
01/18/2022 19:38:10
yes yes yes you're right otherwise an inverse function could never be surjective
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help-10
f4b6863439da4dd896e5804fe0ce0aab
244,129,917,732,651,000
Deus_Vult
01/18/2022 19:38:17
my mistake matas
0
1
null
null
null
null
null
142
help-10
f4b6863439da4dd896e5804fe0ce0aab
730,028,793,875,398,700
Blaxapate
01/18/2022 19:38:38
yes
0
1
null
null
null
null
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help-10
f4b6863439da4dd896e5804fe0ce0aab
244,129,917,732,651,000
Deus_Vult
01/18/2022 19:38:43
a bijective function imposes equal cardinalities of both sets
0
1
730,028,793,875,398,700
Blaxapate
3807
Blaxapate
false
142
help-10
f4b6863439da4dd896e5804fe0ce0aab
565,602,777,275,826,200
Matas
01/18/2022 19:38:45
its fine, this is helping 🙂
1
0
null
null
null
null
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142
help-10
f4b6863439da4dd896e5804fe0ce0aab
565,602,777,275,826,200
Matas
01/18/2022 19:38:48
ok
1
0
565,602,777,275,826,200
Matas
3225
Matas
false
142
help-10
f4b6863439da4dd896e5804fe0ce0aab
565,602,777,275,826,200
Matas
01/18/2022 19:39:01
let me draw 2 functions
1
0
null
null
null
null
null
142
help-10
f4b6863439da4dd896e5804fe0ce0aab
730,028,793,875,398,700
Blaxapate
01/18/2022 19:39:27
(to be pedantic: it's even actually how "having the same cardinality" is formaly defined)
0
1
null
null
null
null
null
142
help-10
f4b6863439da4dd896e5804fe0ce0aab
565,602,777,275,826,200
Matas
01/18/2022 19:40:09
this is bijective right
https://cdn.discordapp.c…2650/unknown.png
unknown.png
1
0
null
null
null
null
null
142
help-10
f4b6863439da4dd896e5804fe0ce0aab
244,129,917,732,651,000
Deus_Vult
01/18/2022 19:40:51
fair point heh
0
1
null
null
null
null
null
142
help-10
f4b6863439da4dd896e5804fe0ce0aab
244,129,917,732,651,000
Deus_Vult
01/18/2022 19:40:56
yes
0
1
244,129,917,732,651,000
Deus_Vult
9771
Deus_Vult
false
142
help-10
f4b6863439da4dd896e5804fe0ce0aab
565,602,777,275,826,200
Matas
01/18/2022 19:42:19
and this is not injective as 0 and 2 map to 1 and also not surjective as nothing maps to 2
https://cdn.discordapp.c…2024/unknown.png
unknown.png
1
0
null
null
null
null
null
142
help-10
f4b6863439da4dd896e5804fe0ce0aab
244,129,917,732,651,000
Deus_Vult
01/18/2022 19:42:40
Exactly right.
0
1
730,028,793,875,398,700
Blaxapate
3807
Blaxapate
false
142
help-10
f4b6863439da4dd896e5804fe0ce0aab
565,602,777,275,826,200
Matas
01/18/2022 19:42:45
thus also not bijective
1
0
565,602,777,275,826,200
Matas
3225
Matas
false
142
help-10
f4b6863439da4dd896e5804fe0ce0aab
565,602,777,275,826,200
Matas
01/18/2022 19:42:57
ok, thank you very much guys :))
1
0
null
null
null
null
null
142
help-10
f4b6863439da4dd896e5804fe0ce0aab
730,028,793,875,398,700
Blaxapate
01/18/2022 19:45:03
no problem 🙂
0
1
565,602,777,275,826,200
Matas
3225
Matas
false
142
help-10
f4b6863439da4dd896e5804fe0ce0aab
244,129,917,732,651,000
Deus_Vult
01/18/2022 19:52:48
you're welcome ! and thanks @Blaxapate
0
1
null
null
null
null
null
142
help-10
f4b6863439da4dd896e5804fe0ce0aab
244,129,917,732,651,000
Deus_Vult
01/18/2022 19:53:11
_if you're willing you can help me out on help-15 channel_
0
1
null
null
null
null
null
143
help-10
b923cada9b084e889cb2cae8f1b23fc1
456,226,577,798,135,800
Deleted User
01/18/2022 20:43:33
Someone ping 1mbibe please? Dude just blocked me and i have no clue why
1
0
null
null
null
null
null
143
help-10
b923cada9b084e889cb2cae8f1b23fc1
456,226,577,798,135,800
Deleted User
01/18/2022 20:43:35
Xd
1
0
730,028,793,875,398,700
Blaxapate
3807
Blaxapate
false
143
help-10
b923cada9b084e889cb2cae8f1b23fc1
701,895,792,763,600,900
s_chanch
01/18/2022 20:44:27
?????
0
1
null
null
null
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null