# Correct the definition for theses binary operations

#### Jonroberts74

I have to correct the italicized word in the following there statements.

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

#### HallsofIvy

MHF Helper
They are all perfectly correct. Yes, in the first it would be better if, like the other two, it said "on S" and "for all $$\displaystyle a, b\in S$$" but I wouldn't consider that a 'deal breaker'.

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