Let A be a nonempty subset of R that is bounded below. Define the set B={b: b is a lower bound of A}. Show that sup B inf A. %3D
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- 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. 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 .
- Suppose thatis an onto mapping from to. Prove that if ℒ, is a partition of, then ℒ, is a partition 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.Label each of the following statements as either true or false. Every upper bound of a nonempty set S must be an element of S.
- Let (A) be the power set of the nonempty set A, and let C denote a fixed subset of A. Define R on (A) by xRy if and only if xC=yC. Prove that R is an equivalence relation on (A).29. Suppose , , represents a partition of the nonempty set A. Define R on A by if and only if there is a subset such that . Prove that R is an equivalence relation on A and that the equivalence classes of R are the subsets .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.
- 16. Let and define on by if and only if . Determine whether is reflexive, symmetric, or transitive.13. Consider the set of all nonempty subsets of . Determine whether the given relation on is reflexive, symmetric or transitive. Justify your answers. a. if and only if is subset of . b. if and only if is a proper subset of . c. if and only if and have the same number of elements.Label each of the following statements as either true or false. Let f:AB where A and B are nonempty. Then f1(f(T))=T for every subset T of B.