Solution for Section G.3 Question 1

1. We must check that the following properties hold.

For (Q,+):

• closure
  Since the sum of any two rational numbers is rational, Q is closed under addition.
  
• associativity
  Addition is associative.
  
• commutativity
  Addition is commutative.
  
• identity
  The rational number 0 is the (additive) identity, since 0 + r = 0 for all rational numbers r.
  
• inverse
  The rational number -r is the (additive) inverse of the rational number r, since -r + r = 0.
  

For (Q-{0}, ·):

• closure
  Since the product of any two non-zero rational numbers is a non-zero rational number, Q-{0} is closed under multiplication.
  
• associativity
  Multiplication is associative.
  
• commutativity
  Multiplication is commutative.
  
• identity
  The rational number 1 is the (multiplicative) identity, since 1 · r = 1 for all rational numbers r.
  
• inverse
  The rational number 1/r is the (multiplicative) inverse of the rational number r, since 1/r · r = 1.
  

For all a,b,c in Q, a · (b + c) = a · b + a · c
The statement is true since we are dealing with ordinary multiplication and addition.

Back to Section G.3