Associative or Commutative of Binary Operation

Question

I need to figure out whether these binary operations are commutative or associative. And then whether a unity exists (but I don't know what that means).

M=$\mathbb{Z}$; a*b=a-b
M=$\mathbb{Q}$; a*b=$\frac{1}{2}$ab

I understand commutative property but not associative for these.

score 2 · Answer 1 · answered Oct 07 '15 at 02:25

2

Commutative means $a*b = b*a$.

Associative means $a*(b*c) = (a*b)*c$.

A unity element $e$ is one such that $a*e = a$ for all $a$ (this is a right unity - a left unity is defined similarly).

So, put these definitions into your operator definitions and see which holds and which do not. For the unity, see what properties the unity element must have.

answered Oct 07 '15 at 02:25

marty cohen

107,799

So for the associative property for ab=a-b, I would have (ab)c=(a-b)c=(a-b)-c=a-b-c a(bc)=a(b-c)=a-(b-c)=a-b+c therefore (ab)c does not equal a(b-c) – ematth7 Oct 08 '15 at 20:53
Correct. Now do the same for everything else. – marty cohen Oct 08 '15 at 21:22

Associative or Commutative of Binary Operation

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