Solution Found!
For each of the following binary relations r on Z, decide
Chapter 5, Problem 2(choose chapter or problem)
For each of the following binary relations \(\rho\) on \(\mathbb{Z}\), decide which of the given ordered pairs belong to \(\rho\).
a. \(x \rho y\) \(\longleftrightarrow\) x \(\lvert\) y; (2, -6), (3, 5), (8, 4), (4, 8)
b. \(x \rho y\) \(\longleftrightarrow\) x and y are are relatively prime; (5, 8), (9, 16), (6, 8), (8, 21)
c. \(x \rho y\) \(\longleftrightarrow\) gcd(x, y) = 7; (28, 14), (7, 7), (10, 5), (21, 14)
d. \(x \rho y\) \(\longleftrightarrow\) \(x^2\) + \(y^2\) = \(z^2\) for some integer \(z\); (1, 0), (3, 9), (2, 2), (-3, 4)
e. \(x \rho y\) \(\longleftrightarrow\) \(x\) is a number from the Fibonacci sequence; (4, 3), (7, 6), (7, 12), (20, 20)
Questions & Answers
QUESTION:
For each of the following binary relations \(\rho\) on \(\mathbb{Z}\), decide which of the given ordered pairs belong to \(\rho\).
a. \(x \rho y\) \(\longleftrightarrow\) x \(\lvert\) y; (2, -6), (3, 5), (8, 4), (4, 8)
b. \(x \rho y\) \(\longleftrightarrow\) x and y are are relatively prime; (5, 8), (9, 16), (6, 8), (8, 21)
c. \(x \rho y\) \(\longleftrightarrow\) gcd(x, y) = 7; (28, 14), (7, 7), (10, 5), (21, 14)
d. \(x \rho y\) \(\longleftrightarrow\) \(x^2\) + \(y^2\) = \(z^2\) for some integer \(z\); (1, 0), (3, 9), (2, 2), (-3, 4)
e. \(x \rho y\) \(\longleftrightarrow\) \(x\) is a number from the Fibonacci sequence; (4, 3), (7, 6), (7, 12), (20, 20)
ANSWER:Step 1 of 6
(a)
Given that \(x\rho y \leftrightarrow x|y; \left( {2, - 6} \right),\left( {3,5} \right),\left( {8,4} \right),\left( {4,8} \right)\) that is \(x\) divides \(y\)
Take the order pair \(\left(2,\ -6\right)\).
\(x=2,\ y=-6)\)
s.t \(x|y,\frac{{ - 6}}{2} = 3)\)
Take the order pair \(\left(3,\ 5\right)\).
\(x = 3,y = 5\)
\(x|y,\frac{5}{3} = 1.67\)
Take the order pair \(\left(8,\ 4\right)\).
\(x = 8,y = 4\)
\(x|y,\frac{4}{8} = 0.5\)
Take the order pair \(\left(4,\ 8\right)\).
\(x = 4,y = 8\)
\(x|y,\frac{8}{4} = 2\)
We can observe that ordered pairs \(\left(2,\ -6\right)\) and \(\left(4,\ 8\right)\) are belongs to relation \(\rho\).