Let \( \mathrm{N}=\{1,2,3,4, \ldots\} \) be the set of natural numbers and \( \mathrm{S}=\{1,4,9,16, \ldots\} \) be the set of squares of the natural numbers. Then \( \mathrm{N} \sim \mathrm{S} \), since we have the one-to-one correspondence \( 1 \leftrightarrow 1,2 \leftrightarrow 4,3 \leftrightarrow 9,4 \leftrightarrow 16, \ldots \mathrm{n} \leftrightarrow \mathrm{n}^{2} \). (This example is interesting, since it shows that an infinite set can be equivalent to a proper subset of itself.) Show that each of the following pairs of sets are equivalent by carefully describing a one-to-one correspondence between the sets. Complete parts (a) through (c) below. A. For each element in \( W \), there is an element in \( N \) that is 1 greater than double that element. B. For each element in \( W \), there is an element in \( N \) that is 1 greater than that element. (a) The whole numbers and natural numbers, \( W=\{0,1,2,3, \ldots\} \) and \( N=\{1,2,3,4, \ldots\} \) C. For each element in \( W \), there is an element in \( N \) that is 2 greater than that element. D. For each element in \( W \), there is an element in \( N \) that is double that element. Write a general rule for the correspondence described in the previous step using \( w \) as an element in \( W \). W.one correspondence between the two sets? \( \square \) (b) The sets of odd and even numbers, \( D=\{1,3,5,7, \ldots\} \) and \( E=\{2,4,6,8, \ldots\} \) Which of the following describes a one-to-one correspondence between the two sets?
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Let’s dive into the fascinating world of set equivalence by exploring some creative correspondences! First, for the pairs of sets representing whole numbers \( W = \{0, 1, 2, 3, \ldots\} \) and natural numbers \( N = \{1, 2, 3, 4, \ldots\} \), we can establish a one-to-one correspondence by using the rule \( n = w + 1 \). Here, each element \( w \) in \( W \) is mapped to an element \( n \) in \( N\) that is simply 1 greater than the element in \( W \). So, for instance, \( 0 \leftrightarrow 1, 1 \leftrightarrow 2, \) and so forth! Now, regarding the sets of odd and even numbers \( D = \{1, 3, 5, 7, \ldots\} \) and \( E = \{2, 4, 6, 8, \ldots\} \), we can set up our correspondence with the rule \( e = d + 1 \). This means for every odd number \( d \) in \( D\), we can find a unique even number \( e \) in \( E\) that is always 1 greater! So, we have \( 1 \leftrightarrow 2, 3 \leftrightarrow 4, \) etc. This delightful pairing shows how both infinite sets beautifully correspond with each other!
