13.3.3. Group of Non-zero Real Numbers under Multiplication

Try these first

1. 1.

   What property ensures that the operation between any two elements of a group results in an element within the set?

   Hint

   Think about the 'closure' of a set.

2. 2.

   What is the identity element when using multiplication in a group of non-zero real numbers?

   Hint

   What number keeps other numbers the same when multiplied?

3. 3.

   Which property guarantees that the product of two elements is still within the set?

   • Closure
   • Identity
   • Inverse

   Hint

   Think about what happens when you add two elements.

4. 4.

   True or False: Every integer has a multiplicative inverse within the set of integers.

   • True
   • False

   Hint

   Consider if dividing integers always yields integers.

5. 5.

   Create a set that fails to meet the group axioms and identify which axiom is violated.

   Hint

   Explore what happens with the operation on elements of this set.

6. 6.

   Prove that addition of integers is a group.

   Hint

   Use the definitions of axioms to support your argument.