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?

Question

UpStudy AI · Accepted Answer

To show that the sets are equivalent, we will establish a one-to-one correspondence between the elements of the sets as described in the problem. Let's go through each part step by step.

### Part (a): The whole numbers and natural numbers
Let $$ W = \{0, 1, 2, 3, \ldots\} $$ and $$ N = \{1, 2, 3, 4, \ldots\} $$.

**Correspondence:**
We can define a function $$ f: W \to N $$ as follows:
$$
f(w) = w + 1
$$
This means:
- For $$ w = 0 $$, $$ f(0) = 0 + 1 = 1 $$
- For $$ w = 1 $$, $$ f(1) = 1 + 1 = 2 $$
- For $$ w = 2 $$, $$ f(2) = 2 + 1 = 3 $$
- For $$ w = 3 $$, $$ f(3) = 3 + 1 = 4 $$
- And so on.

**General Rule:**
The general rule for the correspondence is:
$$
f(w) = w + 1
$$

### Part (b): The sets of odd and even numbers
Let $$ D = \{1, 3, 5, 7, \ldots\} $$ and $$ E = \{2, 4, 6, 8, \ldots\} $$.

**Correspondence:**
We can define a function $$ g: D \to E $$ as follows:
$$
g(d) = d + 1
$$
This means:
- For $$ d = 1 $$, $$ g(1) = 1 + 1 = 2 $$
- For $$ d = 3 $$, $$ g(3) = 3 + 1 = 4 $$
- For $$ d = 5 $$, $$ g(5) = 5 + 1 = 6 $$
- For $$ d = 7 $$, $$ g(7) = 7 + 1 = 8 $$
- And so on.

**General Rule:**
The general rule for the correspondence is:
$$
g(d) = d + 1
$$

### Summary of Correspondences
- For part (a), the correspondence between $$ W $$ and $$ N $$ is given by $$ f(w) = w + 1 $$.
- For part (b), the correspondence between $$ D $$ and $$ E $$ is given by $$ g(d) = d + 1 $$.

Both pairs of sets are equivalent as we have established a one-to-one correspondence between their elements.
For part (a), each whole number $$ w $$ corresponds to $$ w + 1 $$ in the natural numbers. For part (b), each odd number $$ d $$ corresponds to $$ d + 1 $$ in the even numbers.

Upstudy AI Solution

Answer

Solution

Extra Insights

Related Questions

Latest Pre Algebra Questions