Suppose $R$ is a partial order on $A$ and $S$ is a partial order on $B$. Define a relation $T$ on $A\times B$ such that $(a1, b1) T (a2, b2)$ iff $a_1 R a_2$ and $b_1 S b_2$. Is $T$ a partial order on $A \times B?$
So far:
$R$ is a partial order $A$ and $(a,a)\in R$.
$S$ is a partial order on $B$ and $(b,b)\in S$.
$T$ is a partial order and $((a,b),(a′,b′))\in T$.
In order for $T$ to be a partial order on $A\times B$ I know that it must be reflexive, transitive and anti-symmetric.
I'm a bit stuck on where to go from here though.
While formally speaking relations (and functions!) are sets of ordered pairs, it can be quite cumbersome to write everytime things like $(a,a)\in R$. The best thing to do most of the time is to manipulate $R,S$ the same way as $\leq, \geq, \ldots$.
Anti-simmetry: $(a_1,b_1)T(a_2,b_2) \Rightarrow a_1Ra_2$ and $(a_2,b_2)T(a_1,b_1) \Rightarrow a_2Ra_1$ so $a_1 = a_2$. Same thing with $b_1, b_2$, so $(a_1,b_1)=(a_2,b_2)$.
Transivity: $(a_1,b_1)T(a_2,b_2) \Rightarrow a_1Ra_2$ and $(a_2,b_2)T(a_3,b_3) \Rightarrow a_2Ra_3$, so $a_1Ra_3$. Same thing with $b_1,b_2,b_3$ so $(a_1,b_1)T(a_3,b_3)$.
Reflexivity: $(a,b)T(a,b) \iff aRa$ and $bSb$, which is true.