I would also like to add that Havah (Eve) means source of life, mother of life.
Gen. 5 says "In the day that Elohim created man, in the likeness of Elohim made he him; male and female created he them; and blessed them, and called their name Adam, in the day when they were created."
Their name was Adam, not just the man's. Adam means man. Later, the man names his counterpart 'woman' because she was flesh of his flesh and bone of his bone. She was taken out of him so she was 'woman'. But both were still Adam.
Names in Scripture are not arbitrary but have meaning, so understanding why someone is named something is linked to understand their context. ^_^
...and then only after the fall is the Woman named Eve, as the mother of all living.
Mhmm ^_^ Sorry, didn't mean to make it sound like a correction... Just wanted to add it on.