Prove that the each element of a group g has a unique inverse. that is, if a, b, w e g satisfy then b

Step-by-step explanation:

Say is an element of which might have more than 1 inverse. Let's call them , and . So that has apparently two inverses, and .

This means that and that (where is the identity element of the group, and * is the operation of the group)

But so we could merge those two equations into a single one, getting

And operating both sides by b by the left, we'd get:

Now, remember the operation on any group is associative, meaning we can rearrange the parenthesis to our liking, gettting then:

And since b is the inverse of a, , and so:

(since e is the identity of the group)

So turns out that b and c, which we thought might be two different inverses of a, HAVE to be the same element. Therefore every element of a group has a unique inverse.

d. 24+-0.114

step-by-step explanation:

the question requires us to calculate the approximate 95% confidence interval for the mean number of ounces of ketchup per bottle in the sample given the following statistics;

sample mean = 24

standard deviation, σ = 0.4

sample size , n = 49

the confidence interval for a population mean is calculated using the formula;

sample mean ± z-score*(σ/sqrt(n))

the z-score associated with a 95% confidence interval, from the standard normal tables, is;

1.96

substituting these values into the formula;

24 ± 1.96*(0.4/7)

= 24 ± 0.112

from the alternatives given, choice d is closest to the above expression;