Let X = {1, 2, 3, 4}, Y = {2, 3, 4, 5, 6}, Z = {1, 2, 3}. Whats a function f: X -> Y that is one-to-one but

Question

Let X = {1, 2, 3, 4}, Y = {2, 3, 4, 5, 6}, Z = {1, 2, 3}.
Whats a function f: X -> Y that is one-to-one but not onto and whats a function for X -> Y thats neither.

Answer 1

For a one-to-one function define X-->Y this way: 1->2, 2->3, 3->4, and 4->5. This way, two distinct values from X do not have same function value in Y. Also, since the element "6" from Y is never used the function is not an onto function.

For a function that is neither one-to-one nor onto, define g:X->Y such that g(x) = 2 for any value of X. That function isn't one-to-one because distinct x-values have the same y-value; it is not onto because not every value in Y is used.

Answer 2

f:x->Υ f(x)=x+1 is one-to-one but not onto

f:x->Υ f(x)=2 it is'nt one-to-one , it is'nt onto