1. Define a relation on × ( \\ {0} ) by ( x, y ) ( z, w) if xw = yz. a) Prove this relation is an equivalence relation. b) Give a description of the partition of × ( \\ {0} ) formed by the equivalence classes for this equivalence relation. Note: \\ {0} means that in the ordered pair (x,y), y can not be zero. Solution a) We have to show ˜ is reflexive, symmetric, and transitive. (Reflexive) Let (x,y) be an element of R×(R\\{0}). Clearly xy = xy. Thus (x,y)˜(x,y). Therefore ˜ is reflexive. (Symmetric) Assume (x,y)˜(a,b). Then xb=ya, implying ay=bx. Thus (a,b)˜(x,y). Therefore ˜ is symmetric. (Transitive) Assume (x,y)˜(a,b) and (a,b)˜(c,d). Thus xb=ya and ad=bc. Since d is can not be zero, we can divide it to both sides of the second equation, giving a=(bc)/d. Substituting that to the first equation, xb=ybc/d. Since b can not be zero, we can divide both sides by it, then multiplying both sides by d. Thus xd=yc. Implying (x,y)˜(c,d). Therefore ˜ is transitive. b) The partitions are all the rational numbers and two numbers are in the same partition if they can be reduced to the same fraction. For example, (3,6)˜(4,8) since 3*8 = 6*4 (cross multiplying). They both reduce to (1,2), or 1/2..