1873_solutions

F p q fp f q now suppose that f is one one and that y

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Unformatted text preview: hat f is a function from A onto B and that g : B both of the functions f and g have to be one-one. 5 C and that the function g f is one-one, prove that Solution: To see that f is one-one, suppose that x and t are members of the set A and that t Since g f t gfx we see at once that f t x. fx. Now to see that g is one-one, suppose that u and y are members of the set B and that u y. Using the fact that the function f is onto the set B we choose members t and x of A such that u  f t and y  f x . We see at once that t x and therefore gu gft gfx gy. 15. Suppose that f : A B and that r is the relation defined by r  x, y A A fx fy . a. Prove that r is an equivalence relation in the set A. Since f x  f x whenever x A the relation r is reflexive. Since the condition f x  f y is the same as the condition f y  f x the relation r is symmetric. A similar argument shows that r is transitive. b. Prove that if E is a subset of A that contains precisely one member of each equivalence class of r then the restriction of f to E is a one-one function from E into B. Suppose that E is a subset of A containing precisely one member of each equivalence class of r. Given x and t different members of E we know from the fact that x and t do not lie in the same equivalence class of r that f x f t . Therefore the restriction of f to E is one-one. 16. Suppose that S is a given set and that is a set of functions from S to R. Prove that if for any two members f and g of we define the condition f g to mean that f x g x for every x S, then is a partial order of the set . Given any function f we see from the fact that f x f x for every x S that f f. Therefore the relation in is reflexive and similar arguments can be used to show that is also symmetric and transitive. We omit the details. 50 17. Suppose that S is a given set and that is the set of all functions from S to R. Suppose that the partial order is defined in the set above. Given two members f and g of , prove that there exists a member u of such that the following two conditions hold: a. f u and g u b. Whenever a member h of satisfies f h and g h, we have u h. Suppose that f and g belong to the family . We define ux  fx if g x gx if f x  g x fx It is easy to see that this function u has the desired properties. 18. Given any set S, the identity function i S on S is defined by i S x  x for every x S. Prove that if f is a one-one function from a set A onto a set B then f 1 f  i A and f f 1  i B . The equation f 1 f  i A says that f 1 f x  x for every x A and this assertion is exactly the definition of the function f 1 . The equation f f 1  i B follows similarly. 19. Suppose that f : A B. a. Given that there exists a function g : B A such that g f  i A , what can be said about the functions f and g? If x and t are any members of A for which f x  f t then it follows that xgfx gft t and so the function f is one-one. Given any x A we see from the fact that x  g f x that the function g is onto the set A. b. Give...
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This note was uploaded on 11/26/2012 for the course MATH 2313 taught by Professor Staff during the Fall '08 term at Texas El Paso.

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