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So pretty basic question http://prntscr.com/jcz64c

So I am going through the answer, and I check mine, and I got the first one correct, but I cannot figure for the life in me why the second one fails the two closure axioms help really appreciated. As I cannot see any values which would cause it to fail. http://prntscr.com/jcz6m8

mathlover
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1 Answers1

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It appears that the first "axiom" would be violated, in case $(a+c)(b+d)\not =ab+cd$. In particular, if $ad+bc\not =0$. This is, of course, easy to arrange... (e.g. $a=b=c=d=1$)

The second is also violated when $k^2\not =k$ and $a,b\not=0$... (e.g. $a=b=1$ and $k=2$)