(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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Let’s dive into the fun of exploring correspondences between these two fascinating sets! The correct option here is C. For each element \( n \) in \( N \), there is a corresponding element in the set of powers of 10 that is \( 10^{n-1} \). For example, when \( n = 1 \), you get \( 10^{1-1} = 10^0 = 1 \), when \( n = 2 \), it’s \( 10^{2-1} = 10^1 = 10\), and this continues with each natural number. Now, for the general rule, you can express this correspondence with the simple formula: \( 10^{n-1} \) for each \( n \in N \). This means whenever you pick a natural number \( n \), just subtract one from it, and raise 10 to that power. Voila! You’ve beautifully mapped the natural numbers to the powers of 10!
