Let $s$ be the set of complex numbers of the form $a + bi,$ where $a$ and $b$ are integers. we say that $z \in s$ is a unit if there exists a $w \in s$ such that $zw = 1.$ find the number of units in $s.$

Hanna intends to give her granddaughter, melodee, her antique hat pin. this heirloom has been kept under lock and key in the wall vault in the library of hanna's house in virginia. the hat pin is currently the only item in the vault. when hanna is visiting melodee in connecticut, hanna gives melodee the only key to the vault. melodee is grateful for the present and excitedly accepts. in this situation has there been a completed gift?

Universal containers(us) has an integration with its accounting system that creates tens of thousands of orders inside salesforce in a nightly batch. us wants to add automation that can attempt to match leads and contacts to these orders using the email address field on the insert. us is concerned about the performance of the automation with a large data volume. which tool should uc use to automate this process?

Cosmetic profits. sally is the executive vice president of big name cosmetics company. through important and material, nonpublic information, she learns that the company is soon going to purchase a smaller chain of stores. it is expected that stock in big name cosmetics will rise dramatically at that point. sally immediately buys a number of shares of her company's stock. she also tells her friend alice about the expected purchase of stores. alice wanted to purchase stock in the company but lacked the funds with which to do so. although she did not have the funds in bank a, alice decided to draw a check on bank a and deposit the check in bank b and then proceed to write a check on bank b to cover the purchase of the stock. she hoped that she would have sufficient funds to deposit before the check was presented for payment. of which of the following offenses, if any, is alice guilty of by buying stock?