If both $R$ and $S$ are total order, will $T$ be total order?

Question

Let $R$ be a total order on $A$ and let $S$ be total order on $B$. Define $$T = \left\{\left(\left(a,b\right),\left(a',b'\right)\right) \in \left(A \times B\right) \times \left(A \times B\right) \;\vert\; aRa' \;\mathrm{and}\; bSb'\right\}$$ Is $T$ a total order on $A \times B$?

How do i find counter example of this?

Thanks

Answer 1

Consider $R=S=\{0, 1\}$ with $0 \leq 1$. Then in $R \times S$, $(0,1)$ and $(1, 0)$ are not comparable.