I am currently reading Dummit and Foote's Abstract Algebra, and am having a little confusion over the following excerpt:
Suppose that $*$ is a binary operation on a set $G$ and $H$ is a subset of $G$. If the restriction of $*$ to $H$ is a binary operation on $H$, i.e., for all $a,b \in H$, $a * b \in H$, then $H$ is said to be closed under $*$. Observe that if $*$ is an associative (respectively, commutative) binary operation on $G$ and $*$ restricted to some subset $H$ of $G$ is a binary operation on $H$, then $*$ is automatically associative (respectively, commutative) on $H$ as well.
What does it precisely does it mean to restrict a binary operator to the subset $H$?
Restricting $*$ to the subset $H$ means forgetting (or not caring) what $a*b$ is when $a\notin H$ or $b\notin H$.
In other words, the restricted operator is the function $*':H\times H\to G$ defined by $h_1*'h_2=h_1*h_2$ for all $h_1,h_2\in H$.