1873_solutions

The fact that x and t do not lie in the same

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Unformatted text preview: n that there exists a function h : B A such that f h  i B , what can be said about the functions f and h? This is just part a again. The function f must be onto the set B and the function h must be one-one. c. Given that there exists a function g : B A such that g f  i A and that there exists a function h : B A such that f h  i B , what can be said about the functions f, g and h? Now the function f must be one-one and onto the set B and the functions g and h are one-one from B onto A and, in fact g  h. 20. As in a previous example, we define fa x  x a 1 ax whenever a 1, 1 and x 1, 1 . a. Prove that if a and b belong to 1, 1 then so does the number c  ab . 1  ab Hint: An quick way to do this exercise is to observe that c  f b a . We saw in that earlier example that whenever 1  a  1, the function f a is a one-one function from 1, 1 onto 1, 1 . and that f 1  1 and f 1  1. Therefore 1, 1 . cfba b. Given a and b in 1, 1 and prove that f b f a  f c . Given any x 1, 1 we have c  ab , 1  ab 51 fb fa x xa 1 ax  fb xa 1 ax  1ab a b 1ba a b  1 x 1 ax 1 b b xa 1 ax 1 b xa 1 ax 1 xa 1 ax  b 1  ab x a  b 1  ba a  b x  ax  fc x . x Some Elementary Exercises on Set Equivalence 1. Prove that Z  ß Z. Solution: The function f : Z Z  defined by the equation 2n  1 if n fn  2n 0 if n  0 is a one-one function from Z onto Z  . 2. Prove that 0, 1 ß 0, 1 . Solution: The function defined by the equation 1 n 1 fx if nis an integer and n 1 2 2and x  if x  0 x if x 0Þ 0, 1 1 n n 1 n Zand n 2 is a one-one function from 0, 1 onto 0, 1 . 3. Prove that if a  b then any two of the four intervals a, b , a, b , a, b and a, b are equivalent. Solution: At this stage we already know that 0, 1 ß 0, 1 ß 0, 1 ß 0, 1 . If we define f x  a b a x whenever x 0, 1 then we see that f is a one-one function from 0, 1 onto a, b and, in a similar way, we can see that 0, 1 ß a, b and 0, 1 ß a, b and 0, 1 ß a, b . 4. Given that A and B are sets that are disjoint from each other and that A ß Z  and B ß Z  , prove that A Þ B ß Z. Solution: Choose a one-one function f from A onto Z  and a one-one function g from B onto Z  . We now define a function h on the set A Þ B as follows: hx  2f x 1 if x A 2g x if x B and we observe that h is a on-one function from A Þ B onto Z  . 5. Given that A ß B, prove that A A ß B B. Using the fact that A ß B we choose a one-one function f from A onto B. We now define g x, y  f x , f y 52 for every point x, y A B. To see that g is one-one, suppose that x, y and u, v belong to A A and that g x, y  g u, v . We have f x , f y  f u , f v and so f x  f u and f y  f v . Since f is one-one we have x  u and y  v which tells us that x, y  u, v . Now to see that g is onto the set B B, suppose that s, t B B. Using the fact that f is onto B, choose x and y in A such that f x  s and f y  t. We see that g x, y  s, t . Exercises on Binary Operations 1. Prove that if we define x y  x  y xy for all numbers x a...
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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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