(b) Show that the 'rule' g:Z_6 rightarrow Z_9 defined by f([a]_6)=[4a]_9 is not a well-defined function.

Ishaan Booker 2022-07-15 Answered
I'm doing a review for my discrete math test on functions and I'm having troubles with a few questions. Can I get some guidance in how to do these questions so I can be more prepared for the test?
1. (b) Show that the 'rule' g : Z 6 Z 9 defined by f ( [ a ] 6 ) = [ 4 a ] 9 is not a well-defined function.
2. Define a function f : N × N N by f ( ( a , b ) ) = gcd ( a , b )
(a) show that f is not one-to-one
(b) show that f is onto
3. Let A, B, C be non-empty sets and let f : A B and g : B C be functions.
(a) Show that it g f is onto, then g is onto
(b) Find an example of functions f and g such that g f is onto but where f is not onto
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Answers (1)

Carassial3
Answered 2022-07-16 Author has 9 answers
Step 1
2b) I assume the notation [ a ] 6 = a mod 6. Then the function is not well defended because if it was well defined then it should give the same answer when you take a different representative of the equivalent class. Then note f ( [ 1 ] 6 ) = [ 4 ] 9 = 4 but on the other side f ( [ 7 ] 6 ) = [ 28 ] 9 = 1. So the map is not well defined.
1a) It is not one-to-one see f ( 6 , 8 ) = 2 = f ( 10 , 12 ).
1b) Look at f ( n , n ) = n so from this you can conclude that it is onto.
2a) If g f is onto then g is onto on the image of f therefore it is also onto on B, thus g is onto.
2b) If you would have A = B = R and C = x (just one point) then let f = cos ( x ) clearly not onto. And let g = x, (the constant function) then g f is onto but f isn't.

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