In 9-33, determine whether the given relation is reflexive symmetric, transitive, or none of these. Justify your answer.
F is the congruence modulo 5 relation on Z: For every .
To justify whether the given relation is reflexive, symmetric, transitive, or none of these.
F is the congruence modulo 5 relation on Z:
For all m, n ∈ Z, m F n ⇔
Let us consider the given relation as
The relation F is reflective if for every element
is reflexive if it contains for all
We note that F contains and 5 divides 0 (as with 0 an integer) and thus F is reflexive.
The relation F on a set A is symmetric if .
Let us assume that . By definition of F :
By the definition of divides, there exists an integer c such that:
Multiply each side by − 1:
This last inequality then implies that is an integer (as c is an integer), which implies and thus F is symmetric
Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!Get Started