19.1. Discrete Mathematics

Try these first

1. 1.

   Define a ring in your own words.

   Hint

   Think about the combination of a set and operations.

2. 2.

   What is an example of a ring?

   Hint

   Consider commonly used number sets.

3. 3.

   What are the properties required for a ring?

   • Closure
   • Associativity
   • Identity
   • Closure
   • Commutativity
   • Inverses
   • Associativity
   • Existence of Elements

   Hint

   Recall the axioms discussed in class.

4. 4.

   All non-zero elements in a field have an inverse.

   • True
   • False

   Hint

   Think about what makes fields special compared to rings.

5. 5.

   Given a ring of integers mod 5, determine whether the number 2 has a multiplicative inverse.

   Hint

   Try finding a number that gives 1 when multiplied by 2.

6. 6.

   Prove that the set of polynomials over a ring forms a ring itself.

   Hint

   Focus on how addition and multiplication of coefficients work in the original ring.