Question

Can we consider the null set (with any operation) a group?

Answer 1

@hero @Directrix

Answer 2

sorry for tagging you but I figure you could give me a quick exlpination

Answer 3

@zzr0ck3r Any group with elements A,B,C.. must satisfy four properties under a binary operation: closure, associativity, identity, and invertibility.

The requirements for a set to be a group therefore imply that there must be atleast 1 element and therefore the null set can't be a group.

Answer 4

well I have some problems with the definitions
Definition of a binary operator, says that if a,b are in S then a*b is in S
so
the empty set passes this

if a,b are in S then a*b is in S
this is always true because the antecedent is false

then a group is a set with a binary operator that s.t.
for all a,b,c is S, we have
1) associative
2) identity
3) inverse

now for all means, if a,b,c are in S we have ......
again this is a true statement for the null set, because the antecedent is false....

Answer 5

Note how it says: "if "A" and "B" are in S"... But the null set has no elements.

Answer 6

exactly
so a cant be S and thus the antecedent is false so the statement is true

Answer 7

a implies b is always true when a is false

Answer 8

there are strange things that happen with the null set under definitions
example
any metric induced on the null set forms a open and closed set.

Answer 9

so \(\emptyset\) is both open and closed.

Answer 10

anyway...

Answer 11

if a pig can fly then the earth is the center of the universe

true statement ^^^^

Answer 12

the null set is open and closed? hmmm. that kind of makes sense lol.

and why would a pig fly? oh i see what you mean! like if u carry a pig on an airplane..

@zzr0ck3r

Answer 13

no the pig cant fly and thus the statement is true

Answer 14

dafuq

Answer 15

its logic
a implies b is only false when a is true and b is false

Answer 16

so {} is open and closed
yet the definition of a open set, is

if a is in A then there exists open ball at a s.t. the open ball is in A
a is never in A, so thus we have a true statement and the null set is thus open...

so this is why im asking about the group

Answer 17

ya the logic here is not making sense. a => b only when a is false. that's it.

Answer 18

btw, who gives a sht lol.

Answer 19

grrr i said that backwards

Answer 20

no its not that a implies b only when a is false its
the statement is
a implies b is false only if a is true and b is false

Answer 21

oh so you're starting off with the assumption that the aforementioned statement is true to start off with?

Answer 22

im starting with a false assumption, thus the implication is true no matter what.

http://en.wikipedia.org/wiki/Truth_table
scroll down to logical implication

Answer 23

stfu ik what ur saying

Answer 24

oh lol

Answer 25

ok:)

Answer 26

I think its a stupid rule, and this is why.

Answer 27

lol k

Answer 28

lol

Answer 29

where you been?

Answer 30

idk openstudy was getting boring so i shifted to do some higher level math and to prepare for the upcoming national math olympiad...also i have 3 APs this year alongside so its hard to come on openstudy...lol

Answer 31

yeah I hear you. I was on a bunch in the summer because I was in europe and on long train rides. This will be my hardest term yet, so I only come on here to get you bastards to help me:)

Answer 32

xD

Answer 33

I miss that guy Joe-something he was so good with abstract algebra....he moved to texas to start his phd and thus has no time for os.

Answer 34

holy pellet! u guys are that old? lol...

Answer 35

lmfao i wrote "sht" and replaced it with pellet....

Answer 36

wait how old are u rocky? @zzr0ck3r

Answer 37

over 34

Answer 38

err 34

Answer 39

0.0 thats more than twice my age lol...

Answer 40

I was going to say over 30

Answer 41

fu:P

Answer 42

rofl..

Answer 43

twice as wise:P

Answer 44

jokes

Answer 45

hahahahahaha. nice one. should be posted as the joke of the day on jokes.com. =P

Answer 46

lol

Answer 47

btw i turned 16 last month. u did'nt give me b-day beats but its ok :) @zzr0ck3r

Answer 48

there, you get a medal from an old dude

Answer 49

awww ty for the medal lol.

Answer 50

btw, wtf are u doing on openstudy if you're 34? @zzr0ck3r i mean don't u have more adult things to do lol.

Answer 51

lol I am still in school. I have one more term. I come on here because I like math:p trust me i did plenty of pellet before I found math:)

Answer 52

pellet lmfao.... @zzr0ck3r

Answer 53

:)