Beginning Activity Beginning Activity 1: The United States of America
Recall from Section 5.4 that the Cartesian product of two sets \(A\) and \(B\text{,}\) written \(A \times B\text{,}\) is the set of all ordered pairs \(\left( {a,b} \right)\text{,}\) where \(a \in A\) and \(b \in B\text{.}\) That is, \(A \times B = \left\{ {\left( {a,b} \right) \mid a \in A\text{ and } b \in B} \right\}\text{.}\)
Let \(A\) be the set of all states in the United States and let
For example, since California and Oregon have a land border, we can say that (California, Oregon) \(\in R\) and (Oregon, California) \(\in R\text{.}\) Also, since California and Michigan do not share a land border, (California, Michigan) \(\notin R\) and Michigan, California) \(\notin R\text{.}\)
1.
Use the roster method to specify the elements in each of the following sets:
(a)
\(B = \left\{ {y \in A\left| {\left( {\text{ Michigan, } y} \right) \in R} \right.} \right\}\)
(b)
\(C = \left\{ {x \in A\left| {\left( {x,\text{ Michigan } } \right) \in R} \right.} \right\}\)
(c)
\(D = \left\{ {y \in A\left| {\left( {\text{ Wisconsin, } y} \right) \in R} \right.} \right\}\)
2.
Find two different examples of two ordered pairs, \(\left( {x, y} \right)\) and \(\left( {y, z} \right)\) such that \(\left( {x, y} \right) \in R\text{,}\) \(\left( {y, z} \right) \in R\text{,}\) but \(\left( {x, z} \right)\not \in R\text{,}\) or explain why no such example exists. Based on this, is the following conditional statement true or false?
For all \(x, y, z \in A\text{,}\) if \((x, y) \in R\) and \((y, z) \in R\text{,}\) then \((x, z) \in R\text{.}\)
3.
Is the following conditional statement true or false? Explain. For all \(x, y \in A\text{,}\) if \((x, y) \in R\text{,}\) then \((y, x) \in R\text{.}\)