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6. Let S be a set with a finite number of elements, and let f: S→→ S be a map (a) If fis onto, can f not be 1-1? (b) If fis 1-1, can f not be onto? Do your conclusions remain valid even if S has an infinite number of elements?

6. Let S be a set with a finite number of elements, and let f: S→→ S be a map (a) If fis onto, can f not be 1-1? (b) If fis 1-1, can f not be onto? Do your conclusions remain valid even if S has an infinite number of elements?

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.7: Relations
Problem 22E: A relation R on a nonempty set A is called asymmetric if, for x and y in A, xRy implies yRx. Which...
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(a) Show that La is a one-to-one function from G onto G (that is, a bijection from G to G.) (b) Show that for all a,b in G, LaLb =Lab and show that for all a,b in G, RR, =Rpa- (d) (Here Symg represents the set of all permutations on the set G. Some authors, eg. Dummit & Foote, write Sg for this set of permutations.) Show that the map G - Symg defined by a ?- La is an isomorphism from G into Symg. 6. Let S be a set with a finite number of elements, and let f: S → S be a map (a) If f is onto, can ƒ not be 1-1? (b) If f is 1-1, can ƒ not be onto? (c) Do your conclusions remain valid even if S has an infinite number of elements? 7. Let X be a set and X* the set of all functions from X into X. Show that function composition on X is associative on X*. Then prove or disprove if the composition of injective functions is injective. 8. Let SCR, and recall M = supS is the least upper bound of S. Suppose that S is bounded, so supS E R. (If S was not bounded, then we set sups = c0). Show that for every ɛ> 0, there exists x ES such that |x-M| <ɛ and that there exists a sequence (x,) *n=1 from S such that for each n EN, |Xn - M| < 1/2". Define M to be the supremum of S. R and N are the real numbers and natural numbers, respectively. 3
Transcribed Image Text:(a) Show that La is a one-to-one function from G onto G (that is, a bijection from G to G.) (b) Show that for all a,b in G, LaLb =Lab and show that for all a,b in G, RR, =Rpa- (d) (Here Symg represents the set of all permutations on the set G. Some authors, eg. Dummit & Foote, write Sg for this set of permutations.) Show that the map G - Symg defined by a ?- La is an isomorphism from G into Symg. 6. Let S be a set with a finite number of elements, and let f: S → S be a map (a) If f is onto, can ƒ not be 1-1? (b) If f is 1-1, can ƒ not be onto? (c) Do your conclusions remain valid even if S has an infinite number of elements? 7. Let X be a set and X* the set of all functions from X into X. Show that function composition on X is associative on X*. Then prove or disprove if the composition of injective functions is injective. 8. Let SCR, and recall M = supS is the least upper bound of S. Suppose that S is bounded, so supS E R. (If S was not bounded, then we set sups = c0). Show that for every ɛ> 0, there exists x ES such that |x-M| <ɛ and that there exists a sequence (x,) *n=1 from S such that for each n EN, |Xn - M| < 1/2". Define M to be the supremum of S. R and N are the real numbers and natural numbers, respectively. 3
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