Chapter 3.10, Problem 37WE

a.

To determine

To express: The ordered pair $\left(5,3\right)$ in terms of pairs in three other ways.

Answer to Problem 37WE

Three ways of writing the ordered pair $\left(5,3\right)$ ,

  $\begin{array}{l}\left\{\left\{5,3\right\},5\right\}\\ \left\{5,\left\{5,3\right\}\right\}\\ \left\{5,\left\{3,5\right\}\right\}\end{array}$

Explanation of Solution

Given: A definition of “ordered pair”.: The pair “ $a,b$ ” is the set $\left\{a,b\right\}$

  $\left(\text{which}\text{the}\text{same}\text{as}\left\{b,a\right\}\right)$ . We can define ordered pair in terms of the simple concept of pair: The ordered pair “ $a,b$ ” written $\left(a,b\right)$ , is the pair having as its two members $a$ and the pair “ $a,b$ .” Thus,

  $\left(a,b\right)=\left\{a,\left\{a,b\right\}\right\}\text{or}\left(a,b\right)=\left\{\left\{b,a\right\},a\right\}$ .

By using the definitions given the three ways to express the ordered pair $\left(5,3\right)$ are,

  $\begin{array}{l}\left\{\left\{5,3\right\},5\right\}\to \text{by using }\left(a,b\right)=\left\{\left\{a,b\right\},a\right\}\\ \left\{5,\left\{5,3\right\}\right\}\to \text{by using }\left(a,b\right)=\left\{a,\left\{a,b\right\}\right\}\\ \left\{5,\left\{3,5\right\}\right\}\to \text{by using }\left(a,b\right)=\left\{a,\left\{b,a\right\}\right\}\end{array}$

Thus we can write the given ordered pair $\left(5,3\right)$ in three different ways.

b.

To determine

To find: What ordered pair is $\left\{4,\left\{1,4\right\}\right\}$ ?

Answer to Problem 37WE

The ordered pair of $\left\{4,\left\{1,4\right\}\right\}$ is $\left(4,1\right)$ .

Explanation of Solution

Given: A definition of “ordered pair”.: The pair “ $a,b$ ” is the set $\left\{a,b\right\}$

  $\left(\text{which}\text{the}\text{same}\text{as}\left\{b,a\right\}\right)$ . We can define ordered pair in terms of the simple concept of pair: The ordered pair “ $a,b$ ” written $\left(a,b\right)$ , is the pair having as its two members $a$ and the pair “ $a,b$ .” Thus,

  $\left(a,b\right)=\left\{a,\left\{a,b\right\}\right\}\text{or}\left(a,b\right)=\left\{\left\{b,a\right\},a\right\}$ .

By using the given definitions,

  $\begin{array}{l}\left\{4,\left\{1,4\right\}\right\}\text{is}\text{similar}\text{to}\left(a,b\right)=\left\{a,\left\{b,a\right\}\right\}\\ a=4andb=1\\ \left(4,1\right)\end{array}$

Thus we can find the ordered pair of $\left\{4,\left\{1,4\right\}\right\}$ .

c.

To determine

To write: The definition of the ordered triple $\left(a,b,c\right)$ in terms of pairs.

Answer to Problem 37WE

The ordered triple $\left(a,b,c\right)$ in terms of pairs is $\left\{a,\left\{a,b\right\},\left\{a,b,c\right\}\right\}$ .

Explanation of Solution

Given: A definition of “ordered pair”.: The pair “ $a,b$ ” is the set $\left\{a,b\right\}$ $\left(\text{which}\text{the}\text{same}\text{as}\left\{b,a\right\}\right)$ . We can define ordered pair in terms of the simple concept of pair: The ordered pair “ $a,b$ ” written $\left(a,b\right)$ , is the pair having as its two members $a$ and the pair “ $a,b$ .” Thus,

  $\left(a,b\right)=\left\{a,\left\{a,b\right\}\right\}\text{or}\left(a,b\right)=\left\{\left\{b,a\right\},a\right\}$ .

The definition of the ordered triple $\left(a,b,c\right)$ in terms of pairs is,

  $\left\{a,\left\{a,b\right\},\left\{a,b,c\right\}\right\}$

The above pairs represent the triple $\left(a,b,c\right)$ .