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In mathematics, given two preordered sets A {\displaystyle A} and B , {\displaystyle B,} the product order is a partial ordering on the Cartesian product A × B . {\displaystyle A\times B.} Given two pairs {\displaystyle \left} and {\displaystyle \left} in A × B , {\displaystyle A\times B,} declare that ≤ {\displaystyle \left\leq \left} if and only if a 1 ≤ a 2 {\displaystyle a_{1}\leq a_{2}} and b 1 ≤ b 2 . {\displaystyle b_{1}\leq b_{2}.}