Question

let A= {1,2} and C={8,9}. for each i=1,2, construct sets B sub i as well as functions f sub i: A to B sub I, 1<=i<=4, with the following properties:
1) g sub 1 ° f sub 1 is onto C but f sub 1 is not onto B sub I.
2) g sub 2° f sub 2 is one-to-one but g sub 2 is not one-to-one.

Answer 1

Given that,

,   

Now, we have to construct the sets    as well as the functions such that,

1) is onto on C but is not onto on B.

2) is one-one but is not one-one.

For the solutions, we have to consider

and

then, and is defined.

Now, choose and the function work as follows:

then, is onto on C since

but is not onto on B since 'c' has no pre-image in A.

again, if we choose

then, is one-one since

but is not one-one since .