Let \( N=\{1,2,3,4, \ldots\} \) be the set of natural numbers and \( S=\{1,4,9,16, \ldots\} \) be the set of squares of the natural numbers. Then \( N \sim S \), since we have the one-to-one correspondence \( 1 \leftrightarrow 1,2 \leftrightarrow 4,3 \leftrightarrow 9,4 \leftrightarrow 16, \ldots n \leftrightarrow n^{2} \). (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. 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. \( \square \) (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? A. For each element in \( D \), there is an element in \( E \) that is 3 greater than that element. B. For each element in D , there is an element in E that is 1 greater than that element. F. For each element in D, there is an element in E that is double that element. D. For each element in D , there is an element in E that is 2 less than double that element.
Solución de inteligencia artificial de Upstudy
Responder
Solución
¡Inicia sesión para desbloquear respuestas gratis!
Una plataforma de aprendizaje en la que confían millones de estudiantes y profesores reales.
Descubrir Respondido por UpStudy AI y revisado por un tutor profesional
Mind Expander
To establish a one-to-one correspondence for the sets \( D=\{1,3,5,7,\ldots\} \) (the odd numbers) and \( E=\{2,4,6,8,\ldots\} \) (the even numbers), we can use the following relationship: for each element \( d \) in \( D \), there is an element in \( E \) that is 1 greater than that element. More specifically, we can define the correspondence as follows: if \( d = 2k - 1 \) (where \( k \) is a positive integer), then we can map \( d \) to \( e = 2k \) in \( E \). Hence, the general rule for the correspondence is: For each element \( w \in D \), the corresponding element in \( E \) can be expressed as: \[ e = w + 1. \] From the options provided, the correct description of a one-to-one correspondence is: D. For each element in \( D \), there is an element in \( E \) that is 2 less than double that element.
