(a) Make a list of all the injective maps
Show that none is bijective. (This constitutes a direct proof that a set
(b) How many injective maps
are there? (You can see why one would not wish to try to prove directly that there is no bijective correspondence between these sets.)
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Topology
- Give an example of a relation R on a nonempty set A that is symmetric and transitive, but not reflexive.arrow_forward13. 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.arrow_forwardFind 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.arrow_forward
- 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.arrow_forwardIn Exercises 1324, prove the statements concerning the relation on the set Z of all integers. If 0xy, then x2y2.arrow_forwardA relation R on a nonempty set A is called asymmetric if, for x and y in A, xRy implies yRx. Which of the relations in Exercise 2 areasymmetric? In each of the following parts, a relation R is defined on the set of all integers. Determine in each case whether or not R is reflexive, symmetric, or transitive. Justify your answers. a. xRy if and only if x=2y. b. xRy if and only if x=y. c. xRy if and only if y=xk for some k in . d. xRy if and only if xy. e. xRy if and only if xy. f. xRy if and only if x=|y|. g. xRy if and only if |x||y+1|. h. xRy if and only if xy i. xRy if and only if xy j. xRy if and only if |xy|=1. k. xRy if and only if |xy|1.arrow_forward
- True or False Label each of the following statements as either true or false. 2. Every relation on a nonempty set is as mapping.arrow_forwardProve Theorem 1.40: If is an equivalence relation on the nonempty set , then the distinct equivalence classes of form a partition of .arrow_forwardProve that if f is a permutation on A, then (f1)1=f.arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,