In 9-33, determine whether the given relation is reflexive symmetric, transitive, or none of these. Justify your answer.
Define a relation I on R as follows: For all real numbers x and is irrational.
To justify whether the given relation is reflexive, symmetric, transitive, or none of these.
Define a relation I on R as follows: For all real numbers x and y, x I y ⇔ x - y is irrational.
Let us consider the set below.
The relation I is reflective if for every element .
We note that I does not contain because and 0 is rational (since thus I is not reflexive.
It is not reflexive
The relation I on a set A is symmetric if .
Let us assume that
By definition of I :
Let us assume that b − a is rational, by the definition of rational, there exists integers c and d such that:
Multiply each side by − 1:
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