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The set of all positive integers is the union of two disjoint subsets$:\left \{ f\left ( 1 \right ),f\left ( 2 \right ), \dots, f\left ( n \right ), \dots \right \}$ and $\left \{ g\left ( 1 \right ),g\left ( 2 \right ), \dots, g\left ( n \right ), \dots \right \}$, where $f\left ( 1 \right )< f\left ( 2 \right )< \dots < f \left ( n \right ) \dots,$ and $g\left ( 1 \right )< g \left ( 2 \right )< \dots < g\left ( n \right ) \dots,$ and$g\left ( n \right )= f\left ( f\left ( n \right ) \right )+1$ for all $n \geq 1$. What is the value of $g\left ( 1 \right )?$

1. $0$
2. $2$
3. $1$
4. Cannot be determined

1 vote
1
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1 vote