In 9-33, determine whether the given relation is reflexive symmetric, transitive, or none of these. Justify your answer.
E is the congruence modulo 4 relation on Z: For every
To justify whether the given relation is reflexive, symmetric, transitive, or none of these.
E is the congruence modulo 4 relation on Z:
For all m, n ∈ Z, m E n ∈
Let us consider the given relation as
The relation E is reflective if for every element
E is reflexive if it contains for all
We note that E contains because and 4 divides 0 (as with 0 an integer) and thus E is reflexive.
The relation E on a set A is symmetric if .
Let us assume that by the definition of E :
By the definition of divides, there exists an integer c such that:
Multiply each side by − 1:
This last inequality then implies that since − c is an integer (as c is an integer), which implies and thus E is symmetric
Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!Get Started