Discreete math question

Question

Let R be the relation Z defined by aRb if and only if a^2=b^2. Show that R is an equivalence relation on Z and determine its distinct equivalence classes.

Answer 1

Reflexive: For all x it is true that x^{2}=x^{2}. So x is equivalent to itself.

Symmetric: If x is equivalent to y then x^{2}=y^{2} which means that y is equivalent to x.

Transitive: If x is equivalent to y and y equivalent to z then x^{2}=y^{2}=z^{2} which means that x is equivalent to z.

Hence, the relation is an equivalence relation. The classes are the following [x]={y in Z such that y is equivalent to x which means that x^{2}=y^{2}}={y in Z such that x=y or x=-y}={x,-x}.