Below are two relations on $R.$ For each one, determine (with proof) whether or not it is an equivalence relation. If it is an equivalence relation, describe the partition it induces (i.e., describe what the blocks of the partition look like).
a) Define $x \sim y$ to mean that $xy < 0.$
b) Define $x \sim y$ to mean that $x^2 = y^2.$
I need to tutor someone and don't know how to prove this.
You need to test each one for reflexivity, symmetry and transitivity.
First relation:
It fails for reflexivity since we de not have $1\sim 1$ for $1\cdot1\not<0$
Second relation:
reflexivity: $x\sim x$ since $x^2=x^2$
simmetry: if $x\sim y$ then $y\sim x$ since $x^2=y^2$ clearly implies $y^2=x^2$ since equality is symmetric
transitivity: if $x\sim y$ and $y\sim z$ then $x^2=y^2$ and $y^2=z^2$ which implies $x^2=z^2$ since equality is transitive.