Each of the following subsets $R$ of the $(x, y)$-plane defines a relation on the set $\mathbb{R}$ of real numbers. Determine which of the axioms (transitivity, symmetry, reflexivity) are satisfied:
- the set $\{( s, s) | s \in \mathbb{R}\}$
- the empty set
- the locus $\{ xy + 1 = 0 \}$
- the locus $\{ x^2y - xy^2 - x + y = 0\}$
Can someone help me get started on this? I'm having trouble understanding exactly what this question is asking. Maybe do the first one?
For the first one, let $R = \{(s,s) \; | \; s \in \mathbb{R}\}$ be a relation on $\mathbb{R}$.
Reflexivity: For each $s \in \mathbb{R}$, $(s,s) \in R$, since this is how $R$ was constructed.
Symmetry: If $(x,y) \in R$, then also $(y,x) \in R$ because all elements are of the form $(s,s)$.
Transitivity: If $(x,y) \in R$ and $(y,z) \in R$, then also $(x,z) \in R$, again because all elements of $R$ are of the form $(s,s)$.
Does this help you do the last three?