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Beginning Activity Beginning Activity 1: The United States of America
A4US

Recall from Section 5.4 that the Cartesian product of two sets A and B, written A×B, is the set of all ordered pairs (a,b), where a∈A and b∈B. That is, A×B={(a,b)∣a∈A and b∈B}.

Let A be the set of all states in the United States and let

R={(x,y)∈A×A∣x and y have a land border in common }.

For example, since California and Oregon have a land border, we can say that (California, Oregon) ∈R and (Oregon, California) ∈R. Also, since California and Michigan do not share a land border, (California, Michigan) ∉R and Michigan, California) ∉R.

1.

Use the roster method to specify the elements in each of the following sets:

(a)

B={y∈A|( Michigan, y)∈R}

(b)

C={x∈A|(x, Michigan )∈R}

(c)

D={y∈A|( Wisconsin, y)∈R}

2.

Find two different examples of two ordered pairs, (x,y) and (y,z) such that (x,y)∈R, (y,z)∈R, but (x,z)∉R, or explain why no such example exists. Based on this, is the following conditional statement true or false?

For all x,y,z∈A, if (x,y)∈R and (y,z)∈R, then (x,z)∈R.

3.

Is the following conditional statement true or false? Explain. For all x,y∈A, if (x,y)∈R, then (y,x)∈R.