Let us define a function f:N→Q as follows, f(n)={4n−1, if n is odd2n+1, if n is even.
Since f(7)=47−1=46=23 and f(2)=22+1=23, we conclude that f(7)=f(2), and hence a function f:N→Q is not one-to-one.
Taking into account that 4n−1=81 implies n=21+1=23∈/N and 2n+1=81 implies n=41−1=−43∈/N, we conclude that for 81∈Q the preimage f−1(81)=∅, and thus a function f:N→Q is not onto.
Therefore, the statements "f is not one to one" and "f is not onto" are true, and the staterments "f is onto" and "f is one to one" are false.
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