Define a relation r on NxN by (a,b)r(c,d) iff a+d=b+c (see Definition A.2.4. of the reading assignment for 2/29). Show that r is an equivalence relation, i.e. verify that this relation is reflexive, symmetric, and transitive.
(Note that the operation of subtraction should not be used as it is not defined on N, however, any properties of addition in N can be used as they can be derived from the definition. We have listed a few properties in class.)
Which of the following statements are true? Prove or disprove each statement.
The sum of two rational numbers is always rational.
The sum of two irrational numbers is always irrational.
The sum of a rational number and an irrational number is always irrational.
The product of a rational number and an irrational number is always irrational.
Hints: use the definition of a rational number (the one that is usually taught in high school, i.e. a real number is called rational if it can be written as a quotient of
integers). You may want to use a proof by contradiction for some of these statements. To disprove a statement, give a specific counterexample.