In 9-33, determine whether the given relation is reflexive symmetric, transitive, or none of these. Justify your answer.
Define a relation Q on R as follows: For all real numbers x and is rational.
To justify whether the given relation is reflexive, symmetric, transitive, or none of these.
Define a relation Q on R as follows: For all real numbers x and y, x Q y ⇔ x - y is rational.
Let us consider the set below.
The relation Q is reflexive if for every element .
We note that Q contains because and 0 is rational (since thus Q is reflexive.
It is reflexive
The relation Q on a set R Is symmetric if .
Let us assume that By definition of Q :
By the definition of rational, there exists integers c and d such that:
Multiply each side by − 1:
This last inequality then implies that b − a is irrational since is rational (as − c is an integer and d is an integer), which then implies and thus Q is symmetric
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