1)

A binary operation $*$ is

*commutative*if and only if $a*b=b*a$

well commutative is the right word but seems odd its not saying its a binary operation on a set nor confirming a,b are in the set

2)

A binary operation $*$ on a set $S$ is

*associative*if and only if for all $a,b,c \in S$ we have $(b*c)*a=b*(c*a)$

I see no issues with this one

3)

A subset $H$ of $S$ is

*closed*under a binary operation $*$ on $S$ if and only if $(a*b) \in H$ for all $a,b \in S$

seems wrong, seems like it should say for all $a,b \in H$ but maybe it's correct