(c) The sets of natural numbers and the set of powers of $$ 10, \mathrm{~N}=\{1,2,3,4, \ldots\} $$ and $$ \{10,100,1000, \mathrm{~m}\} $$ Which of the following describes a one-to-one comespondence between the two sets? A. For sach element in N , there is an element in the set of powers of 10 that is 10 times that element. B . For sach element in N , there is an element in the set of powers of 10 that is 10 raised to the power of that element. C. For sach element in N , there is an element in the set of powers of 10 that is 10 raised to the power of one less than that element. D. For each elemem in N , there is an element in the set of powers of 10 that is that element raised to the power of to. Write a general rule for the comespondence described in the previous step using in as an element in N .

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1. The set of natural numbers is given by  
   $$
   \mathrm{N} = \{1, 2, 3, 4, \ldots\}
   $$
   and the set of powers of 10 is  
   $$
   \{10, 100, 1000, \ldots\}.
   $$

2. Option B states:  
   "For each element in $$ \mathrm{N} $$, there is an element in the set of powers of 10 that is 10 raised to the power of that element."  
   This implies the function $$ f: \mathrm{N} \to \{10, 100, 1000, \ldots\} $$ defined by  
   $$
   f(n) = 10^n.
   $$
   Checking this:
   - For $$ n = 1 $$, $$ f(1) = 10^1 = 10 $$.
   - For $$ n = 2 $$, $$ f(2) = 10^2 = 100 $$.
   - For $$ n = 3 $$, $$ f(3) = 10^3 = 1000 $$.
   
   This mapping produces exactly the set of powers of 10 as given.

3. The remaining options do not produce the correct correspondence:
   - Option A multiplies $$ n $$ by 10, which does not yield $$ 10^n $$ (e.g., for $$ n = 2 $$, $$ 10 \times 2 = 20 $$ which is not in the set).
   - Option C uses $$ 10^{n-1} $$; for $$ n = 1 $$, $$ 10^{1-1} = 10^0 = 1 $$ which is not in the set.
   - Option D raises $$ n $$ to the power 10, which does not give the numbers in the set of powers of 10.

4. Therefore, the one-to-one correspondence is given by option B.

5. The general rule for the correspondence can be written as:  
   $$
   f(n) = 10^n, \quad \text{where } n \in \mathrm{N}.
   $$
The one-to-one correspondence between the natural numbers and the set of powers of 10 is described by option B: for each natural number $$ n $$, the corresponding element in the set of powers of 10 is $$ 10^n $$. The general rule is $$ f(n) = 10^n $$ for all $$ n $$ in the natural numbers.

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