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 3/3). Show that r is an equivalence relation,
as stated in that definition, i.e. verify that this relation is reflexive, symmetric, and transitive.
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.