Which of these are partitions of the set Z × Z of ordered

Chapter 13, Problem 45E

(choose chapter or problem)

Which of these are partitions of the set \(\mathbf{Z} \times \mathbf{Z}\) of ordered pairs of integers?

a) the set of pairs (x, y), where x or y is odd; the set of pairs (x, y), where x is even; and the set of pairs (x, y), where y is even

b) the set of pairs (x, y), where both x and y are odd; the set of pairs (x, y), where exactly one of x and y is odd; and the set of pairs (x, y), where both x and y are even

c) the set of pairs (x, y), where x is positive; the set of pairs (x, y), where y is positive; and the set of pairs (x, y), where both x and y are negative

d) the set of pairs (x, y), where 3 | x and 3 | y; the set of pairs (x, y), where 3 | x and 3 ✗ y; the set of pairs (x, y), where 3✗ x and 3 | y; and the set of pairs (x, y), where 3 ✗ x and 3 ✗ y

e) the set of pairs (x, y), where x > 0 and y > 0; the set of pairs (x, y), where x > 0 and \(y \leq 0\); the set of pairs (x, y), where \(x \leq 0\) and y > 0; and the set of pairs (x, y), where \(x \leq 0\) and \(y \leq 0\)

f ) the set of pairs (x, y), where \(x \neq 0\) and \(y \neq 0\); the set of pairs (x, y), where x = 0 and \(y \neq 0\); and the set of pairs (x, y), where \(x \neq 0\) and y = 0

Equation Transcription:

Text Transcription:

Z x Z

y leq 0

x leq 0

x neq 0

y neq 0