In 9—33, determine whether the given relation is reflexive, symmetric, transitive, or none of these. Justify your answers.
11. D is the relation defined on R as follows: For every .
To determine whether the given relation is reflexive, symmetric, transitive, or none of these.
D is the relation defined on R as follows:
For all x, y ∈ R, x D y ⇔
Let us consider the given relation as
The relation D is reflexive if for every element
is reflexive if it contains for all
We note that D contains (as the square of a real number is always nonnegative) and thus D is reflexive.
The relation D on a set A is symmetric if for every elements .
Let us assume that by definition of D :
By commutative property of multiplication:
This last inequality then implies that and thus D is symmetric
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