Let n E w. Suppose that f:n → A is onto A. Prove that A is finite.
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- 4. Let , where is nonempty. Prove that a has left inverse if and only if for every subset of .Let g:AB and f:BC. Prove that f is onto if fg is onto.28. Let where and are nonempty. Prove that has the property that for every subset of if and only if is onto. (Compare with Exercise 15c.) Exercise 15c. c. For this same and show that.
- Suppose thatis an onto mapping from to. Prove that if ℒ, is a partition of, then ℒ, is a partition of.[Type here] 21. Prove that ifand are integral domains, then the direct sum is not an integral domain. [Type here]6. For the given subsets and of Z, let and determine whether is onto and whether it is one-to-one. Justify all negative answers. a. b.
- Prove that the cancellation law for multiplication holds in Z. That is, if xy=xz and x0, then y=z.27. Let , where and are nonempty. Prove that has the property that for every subset of if and only if is one-to-one. (Compare with Exercise 15 b.). 15. b. For the mapping , show that if , then .9. If denotes the unity element in an integral domain prove that for all .