Correct the definition for theses binary operations

Sep 2013
567
27
Portland
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
Apr 2005
20,249
7,909
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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