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If you are a genius help!!

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#1
ArtemisZ083

Im having problems in this question, i have asked my friends and teachers but they dk either, please help!!

Let A = {1,2,3,4}. The number of functions f: S ---> S. Such that f(f(i) = i for all 1<= i <= 4 is?
Im getting the answer 24 (4!) by applying the condition that the function must be bijective as its inverse exists. But the answer is 10

#2
XtraChrxs
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idk if vlr is the place for this

#3
Average_NA_fan
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(-demon1)+flor = winning champs

#4
bonkarfanboy1
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No clue how you mean you got 24.

There are 10 solutions:
The first is just f(i) = i (trivial).

Then there are 6 solutions that only flip 2 numbers such as
f(1) = 2, f(2) = 1, f(3) = 3, f(4) = 4

Finally there are 3 solutions that flip 2 pairs of numbers such as
f(1) = 2, f(2) = 1, f(3) = 4, f(4) = 3

"Flipping" numbers is the only way to have f(f(i)) = i so there are no other solutions.

#5
keep
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explain in valorant terms? here is what i did
https://imgur.com/abluRhm

also who tf uses i as a variable 😭

#7
bonkarfanboy1
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Your solution is wrong although subtly. You do not account for my 3 "double flip" functions but you have accounted for the trivial solution 3 extra times. A function acts on every element in the set (at least here it has to for f(f(i))=i to hold).

Here are my 10 solutions: https://imgur.com/a/WleLcVC

#14
ArtemisZ083
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basically 4c1, 4c2 and then 3c1

#10
ArtemisZ083
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What kind of function is this bro? What exactly are you counting? You cant count the images one by one and count them as functions. Its lucky you got the answer

#8
ArtemisZ083
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I got 24 because if f(i) is self invertible
it should be bijective(one one and onto) Before this question i didnt know applying this condition leads to overcounting, thanks for you help.

#6
PP12123213123
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tell me why u put 4! in the answer

(i) f(a) = a, f(b) = b, f(c) = c, f(d) = d ; 1 case
(ii) f(a) = b, f(b) = a, f(c) = c, f(d) = d ; 6 case
(iii) f(a) = b, f(b) = a, f(c) = d, f(d) = c ; 3 case

1+6+3 = 10 case

#11
ArtemisZ083
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I put 4! because every self invertible function should be bijective( one one and onto) so i applied its condition and got 4!. I didnt realise it would lead to overcounting

#9
trikecycle
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1+2+3+4=10

#12
Hades_Loves_Rb
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so bassicly then you and then from there you get and thats that! hope this helped

#13
ArtemisZ083
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It definitely did, thank you puchan

#15
HenBabyH
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Yo, i’m what you would call a Genius

#16
ArtemisZ083
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Who gave the solution of einstein's field eqns

#17
HenBabyH
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Isaac Newton

#18
ArtemisZ083
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Name another scientist except newton and einstein( you are wrong)

#19
HenBabyH
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Curie

#21
ArtemisZ083
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What is the value of K in coulombs law

#23
HenBabyH
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Krypton, next question

#20
ArtemisZ083
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.

#22
HenBabyH
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🦦

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