Let Z* = Z – {0} and define Q to be the quotient set of Z x Z* by the following equivalence relation: (a1,b1) ~ (a2,b2) a1b2 = b1a2. The elements of Q (i.e. the rational numbers) are equivalence classes [(a,b)]~. These are usually denoted by a/b and one calls (a,b) a representative of the rational number a/b. Moreover, a/b can always be written in lowest terms, that is a and b are coprime, or relatively prime. For instance (1,2) and (-3,-6) are distinct representatives of the same rational number ½. Check if the function f : Q X Q - Q f (a1/b1, a2/b2) = (a1+a2)/(b1+b2), is well define

Question

Let Z* = Z – {0} and define Q to be the quotient set of Z x Z* by the following equivalence relation: (a1,b1) ~ (a2,b2) <-> a1b2 = b1a2.
The elements of Q (i.e. the rational numbers) are equivalence classes [(a,b)]~. These are usually denoted by a/b and one calls (a,b) a representative of the rational number a/b. Moreover, a/b can always be written in lowest terms, that is a and b are coprime, or relatively prime. For instance (1,2) and (-3,-6) are distinct representatives of the same rational number ½.
Check if the function f : Q X Q -> Q
f (a1/b1, a2/b2) = (a1+a2)/(b1+b2),
is well define

anonymous · Answer

Q = rational numbers
Z = integers