Let M be a finite metric space. Prove that every set XCM is open and closed.
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- Suppose f,g and h are all mappings of a set A into itself. a. Prove that if g is onto and fg=hg, then f=h. b. Prove that if f is one-to-one and fg=fh, then g=h.Label each of the following statements as either true or false. A mapping is onto if and only if its codomain and range are equal.Label each of the following statements as either true or false. Every epimorphism is an endomorphism.
- Label each of the following statements as either true or false. The least upper bound of a nonempty set S is unique.Suppose thatis an onto mapping from to. Prove that if ℒ, is a partition of, then ℒ, is a partition of.Label each of the following statements as either true or false. Every upper bound of a nonempty set is a least upper bound.
- Give an example of a relation R on a nonempty set A that is symmetric and transitive, but not reflexive.Label each of the following statements as either true or false. 9. Composition of mappings is an associative operation.Find mappings f,g and h of a set A into itself such that fg=hg and fh. Find mappings f,g and h of a set A into itself such that fg=fh and gh.
- Label each of the following statements as either true or false. The Well-Ordering Theorem implies that the set of even integers contains a least element.Label each of the following statements as either true or false. Every least upper bound of a nonempty set S is an upper bound.If A is a compact subset of a metric space (X, d) and B is a closed subset of A, prove that B is also compact.