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.

the correct answer is b. negative exponential growth.

step-by-step explanation:

you can tell it's an exponential function because it decreases rapidly then levels out and never touches the x-axis.

it is negative because the y-axis decreases as the x-axis increases.

therefore it is a negative exponential growth graph.

hope i !

a

step-by-step explanation:

when a number is between two straight lines it means the absolute value.

to find absolute value you need to know how many numbers is that number away from 0.

when finding absolute value the number will always come out positive

so absolute value of -3 is 3

absolute value of 3/2 is 3/2

absolute value of -2 is 2

absolute value of 3.5 is 3.5

absolute value of -1 is 1

now put these numbers in order:

a)   |-1|, |3/2|, |-2|, |-3|, |3.5|