If R1 and R2 are 2 equivalence relations on set S, then Prove that R1 U R2 is reflexive, symmetric but needn't be transitive.
This is the question and I understand why it needn't be transitive but the problem is how do I write it on a paper so that the questioner or examiner can understand that I understand ?
Let $S=\{x,y,z\}$. Besides the reflexive relations $k=k$ for $k\in S$, $R_1$ also has the equivalence $x=y$ while $R_2$ also has the equivalence $y=z$. Then in $R_1\cup R_2$, $x=y$ and $y=z$ but $x\ne z$.
The union of two equivalence relations over the same set is always reflexive and symmetric: any relation $a=a$ or $a=b$ in the union must come from at least one of the component equivalences, and for $a=b$ there will always be a $b=a$ in the same equivalence $a=b$ was found in.