13.2.4. Inverse Element

Try these first

1. 1.

   What is the identity element of integers under addition?

   Hint

   Think of what number when added keeps other numbers the same.

2. 2.

   Define the closure property in your own words.

   Hint

   Consider two numbers from a set and their result.

3. 3.

   What defines the inverse of an element in a group?

   Hint

   Think about what happens mathematically with operations.

4. 4.

   True or False: The set of all integers with addition forms a group.

   • True
   • False

   Hint

   Consider closure, identity, and inverses.

5. 5.

   Show that the set of integers under addition exhibits the properties of an abelian group.

   Hint

   Recall the axiom definitions.

6. 6.

   Find a set that has closure and an identity but no inverses, and justify.

   Hint

   Focus on defining the properties separately.