Homework 5

• 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 2/26). Show that r is an equivalence relation, as stated in that definition, i.e. verify that this relation is reflexive, symmetric, and transitive.

• Show that the operations addition and multiplication given in A.2.4. are well-defined operations on Z, i.e. the result does not depend on the representative from each equivalence class. In other words, if pairs (a,b) and (k,l) are equivalent and pairs (c,d) and (m,n) are equivalent, then (a,b)+(c,d) and (k,l)+(m,n) are equivalent and (a,b)(c,d) and (k,l)(m,n) are equivalent.