13.2.1. Closure Property

Try these first

1. 1.

   What does the closure property state?

   Hint

   Think about operations like addition or multiplication.

2. 2.

   Give an example of a set that satisfies closure under addition.

   Hint

   Consider what happens when you add two integers together.

3. 3.

   What is required for a set to satisfy the closure property?

   • The result must be outside the set.
   • The result must be an element of the set.
   • The set must be finite.

   Hint

   Think about the definition of closure.

4. 4.

   True or False: The set of integers under subtraction is a group.

   • True
   • False

   Hint

   Think about what happens when you subtract.

5. 5.

   Prove that the set of all even integers under addition forms a group and determine its closure.

   Hint

   Start with defining even integers mathematically.

6. 6.

   Consider three operations: +, ×, and modulo. Can a set be closed under one operation but not the others? Provide an example.

   Hint

   Examine closure concerning modular arithmetic.