Classify the following relations as reflexive, irreflexive, symmetric, antisymmetric or transitive. Explain each property in the context of the question. “is greater than or equal to” on the set of real numbers.
My question is to classify the relations do I keep the values for $x, y$ and $z$ the same for each relation. For example say the set is $\{x,y,z\}$ which can be $\{1,2,3\}$ respectively? If so am I correct in the following?
Reflexive: $xRx$ eg. $1 \geq 1$, $2 \geq 2$ and since this is true, $R$ must be reflexive.
Symmetric:
If $xRy$ then $yRx$. If $x = 1$ and $y = 2$ then $1>=2$ it is false so it is not symmetric Is this correct?
Antisymmetric if $xRy$ and $yRx$ then $x=y$
if $xRy$ and $yRx$ then $x=y$, this can only stand true on the condition that $x=y$ so on this condition it is Antisymmetric. Is this correct?
Transitive: If $xRy$ and $yRz$ then $xRz$ It is not transitive as $1 \geq 2 $ and $2 \geq 3$ and $1 \geq 3$ are all false.