13.3.2. Non-negative Integers under Addition

Try these first

1. 1.

   Define the closure property in the context of group theory.

   Hint

   Think about what happens when you add two elements in a set.

2. 2.

   What is the identity element for addition?

   Hint

   Consider the effect of adding zero to a number.

3. 3.

   What does the closure property ensure in a group?

   • The result is always zero
   • The result is an element of the set
   • The result is always greater than zero

   Hint

   Think about adding elements and their results.

4. 4.

   True or False: Every set that includes zero and is closed under addition is a group.

   • True
   • False

   Hint

   Consider the necessity of having inverses in the group definition.

5. 5.

   Given the set of even integers, do they form a group under addition? Justify your answer based on the group properties.

   Hint

   Check each property to confirm.

6. 6.

   Design a new set and operation that fails to meet one of the group properties and identify which property is violated.

   Hint

   Think about operations that might produce results outside the defined set.