英语翻译数学集合论问题(英语)LetZ2bethesetofallpairsofintegers.DefineadditionandmultiplicationonZ2by(m,n)·(p,q):=(m·p,n·q)and(m,n)+(p,q):=(q·m+p·n,nq)(1)for(m,n),(p,q)∈Z2.Definetheequivale

Question

英语翻译
数学集合论问题(英语)
Let Z2 be the set of all pairs of integers.Define addition and multiplication on Z2 by
(m,n) · (p,q) := (m · p,n · q) and (m,n) + (p,q) := (q · m + p · n,nq) (1)
for (m,n),(p,q) ∈ Z2.Define the equivalence relation
(n,m) ∼ (p,q) :⇔ n · q = m · p,(2)
giving rise to the partition Z2/ ∼.
i) Show that the multiplication in (1) can be used to define multiplication on Z2/ ∼ via
[(m,n)] · [(p,q)] := [(m · p,n · q)] (3)
for classes [(m,n)],[(p,q)] ∈ Z2/ ∼.In particular,you need to show that this multiplication is well-defined,
i.e.,it doesn’t depend on the representatives of the classes.In other words,if (m,n),(m′,n′) ∈ [(m,n)]
and (p,q),(p′,q′) ∈ [(p,q)],then
[(m · p,n · q)] = [(m′ · p′,n′ · q′)] that is (m · p,n · q) ∼ (m′ · p′,n′ · q′).(4)
ii) Do the same for the addition in (1).
你能把字母大正确点吗？
回答好的我追30分

Answer 1

整数对、划分、等价关系就不用我来解释了吧?式子末尾括号数字应该是编号 Z2为所有整数对的集合.定义Z2上的加法运算和乘法运算为：(m,n) • (p,q) := (m • p,n • q) and (m,n) + (p,q) := (q •...