Group Under Addition or Multiplication?: The set of numbers of the form 3n, where n ∈ Z. Question as in available in description below (photo).
Would appreciate any help, I'm really finding this challenging.
Would appreciate any help, I'm really finding this challenging.
1 Answer
Mar 4, 2018
Explanation:
A group
#S# is closed under#@# (#a, b in S rarr a@b in S# )#@# is associative (#a, b, c in S rarr (a@b)@c = a@(b@c)# )- There is an identity
#e in S# (#a in S rarr a@e = e@a = a# ) - Every element has an inverse (
#a in S rarr EE b in S : a@b = b@a = e# )
Note that
What about
- If
#3m, 3n in 3ZZ# then#3m + 3m = 3(m+n) in 3ZZ# . So#3ZZ# is closed under#+# #+# is associative, since it is associative in#ZZ# #0 = 3 * 0 in 3ZZ# . So#3ZZ# contains an identity for#+# - If
#3m in 3ZZ# then#3(-m) in 3ZZ# and#3m+3(-m) = 0# . So every element has an inverse.
So
What about
- If
#3m, 3n in 3ZZ# then#(3m) * (3n) = 3(3mn) in 3ZZ# . So#3ZZ# is closed under#*# #*# is associative, since it is associative in#ZZ# #1# is not divisible by#3# so#1 !in 3ZZ# . Hence#3ZZ# contains no identity for#*# #3ZZ# lacks multiplicative inverses. It does not even have an identity,
So