Intersection of equivalence relations ($R={\bigcap }_j{R}_j$)

The intersection of any family of equivalence relations on one set is an equivalence relation. Reflexivity and symmetry hold in every component. A pair of consecutive related points belongs to every component, so each component's transitivity supplies the required final pair. Its equivalence classes refine the component partitions. The empty intersection is the universal relation.

For positive integers $a,b$, partition the nonnegative integers by equality of $\lfloor n/a\rfloor$ and separately by equality of $\lfloor n/b\rfloor$. The intersection of equivalence relations gives nonempty intersections of these consecutive blocks. Its boundaries occur at multiples of either $a$ or $b$.

Let $\ell =\mathrm{lcm}(a,b)$. Translating by $\ell$ increases the two floor quotients by the fixed integers $\ell /a$ and $\ell /b$, so it preserves all within-block equalities and all block boundaries. Thus the refinement repeats in intervals of length $\ell$. If $0=c_0<c_1<\cdots <c_r=\ell$ lists the multiples of $a$ or $b$ within one period, its blocks are exactly $\{j\ell +c_i,\dots ,j\ell +c_{i+1}-1\}$ for $j\ge 0$ and $0\le i<r$. The pattern repeats, but blocks in different periods remain distinct equivalence classes.