Let S be a non-empty open connected subset of R². The number of possible continuous onto functions from S to the set of all rational numbers Q is 02 Oo 0 1 00
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- 3. Let be an integral domain with positive characteristic. Prove that all nonzero elements of have the same additive order .Suppose thatis an onto mapping from to. Prove that if ℒ, is a partition of, then ℒ, is a partition of.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.
- 16. Let and define on by if and only if . Determine whether is reflexive, symmetric, or transitive.Let g:AB and f:BC. Prove that f is onto if fg is onto.6. Prove that if is any element of an ordered integral domain then there exists an element such that . (Thus has no greatest element, and no finite integral domain can be an ordered integral domain.)
- [Type here] 21. Prove that ifand are integral domains, then the direct sum is not an integral domain. [Type here]21. A relation on a nonempty set is called irreflexive if for all. Which of the relations in Exercise 2 are irreflexive? 2. In each of the following parts, a relation is defined on the set of all integers. Determine in each case whether or not is reflexive, symmetric, or transitive. Justify your answers. a. if and only if b. if and only if c. if and only if for some in . d. if and only if e. if and only if f. if and only if g. if and only if h. if and only if i. if and only if j. if and only if. k. if and only if.Prove that if a subring R of an integral domain D contains the unity element of D, then R is an integral domain. [Type here][Type here]