Show that R is an order and draw it’s diagram

Question

Let A = {1,2,3,4}, and let R be a binary relation on A×A given by: ((a,b),(c,d))∈R if and only if a≤c and b≤d. Show that R is an order and draw its diagram.

So I drew the diagram but the part I’m confused about is showing that it’s an order.

Answer 1

To show $R$ is an (partial) order, you need to show it is reflexive, transitive, and anti-symmetric. It is clearly reflexive since $a\leq a$ and $b\leq b$. It is transitive since $(a_1,b_1)\leq (a_2,b_2)$ and $(a_2,b_2)\leq (a_3,b_3)$ imply $a_1\leq a_3$ and $b_1\leq b_3$. If $(a,b)\leq (c,d)$ and $(c,d)\leq (a,b)$, then $a\leq c$ and $c\leq a$, so $a=c$. Similarly with $b$ and $d$, so it must follow $(a,b)=(c,d)$, so $R$ is a partial ordering on $A\times A$.

Answer 2

More generally, if you have two partial orderings $(P_1, \leq_1)$ and $(P_2,\leq_2)$, as it is your case (with $P_1 = P_2 = \{1,2,3,4\}$ and the usual order), then $(P,\leq)$ is also a partial ordering, where $P = P_1 \times P_2$ and $$(a,b) \leq (c,d) \quad\text{ iff } \quad a \leq_1 c \text{ and } b \leq_2 d$$ (which is exactly your case).
Notice that you can even generalize this result from the product of two partial orderings to any number of them (including infinitely many).