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- Let be as described in the proof of Theorem. Give a specific example of a positive element of .For each of the following parts, give an example of a mapping from E to E that satisfies the given conditions. a. one-to-one and onto b. one-to-one and not onto c. onto and not one-to-one d. not one-to-one and not ontoSuppose thatis an onto mapping from to. Prove that if ℒ, is a partition of, then ℒ, is a partition of.
- Let g:AB and f:BC. Prove that f is onto if fg is onto.Label each of the following statements as either true or false. 3. Let where A and B are nonempty. Then for every subset S of A.Suppose X = Rk+, for some k ≥ 2, and define x = (x1,...,xk) ≽ y = (y1,...,yk) if x ≥ y; that is, if for all l = 1, ..., k, xl ≥ yl. Show that ≽ is transitive but not complete
- Prove that Z ≈ E*, where Z is the set of integers and E* is the set of positive even integers.Suppose (X, S) is a measurable space, E1, . . . , En are disjoint subsets of X, and c1, . . . , cn are distinct nonzero real numbers. Prove that c1χE1 + · · · + cnχEn is an S-measurable function if and only if E1, . . . , En ∈ S.Let E be a finite extension of R. Use the fact that C is algebraicallyclosed to prove that E = C or E = R.