Every nonempty class C has an E-minimal element.
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- 3. Let be an integral domain with positive characteristic. Prove that all nonzero elements of have the same additive order .Label each of the following statements as either true or false. Every upper bound of a nonempty set is a least upper bound.Label each of the following statements as either true or false. The least upper bound of a nonempty set S is unique.
- Prove statement d of Theorem 3.9: If G is abelian, (xy)n=xnyn for all integers n.Label each of the following statements as either true or false. Every least upper bound of a nonempty set S is an upper bound.Label each of the following statements as either true or false. If a nonempty set contains an upper bound, then a least upper bound must exist in .
- Let be as described in the proof of Theorem. Give a specific example of a positive element of .Let f:AA, where A is nonempty. Prove that f a has right inverse if and only if f(f1(T))=T for every subset T of A.4. Let , where is nonempty. Prove that a has left inverse if and only if for every subset of .
- Give an example of a relation R on a nonempty set A that is symmetric and transitive, but not reflexive.(See Exercise 26) Let A be an infinite set, and let H be the set of all fS(A) such that f(x)=x for all but a finite number of elements x of A. Prove that H is a subgroup of S(A).23. Prove that if and are normal subgroups of such that , then for all