How to find if neutral/idnty element exists for 16 different binary operations of *?

**x3bnm** · August 24th 2012, 04:49 PM

Suppose I'm given an operation $*$ on a set $A\,\, \text{where}\,\, a,b \in A$ such that the operation between two elements of set $A$ is also an element of set $A$

Now there are $2 \times 2 \times 2 \times 2 = 16$ possible binary operations of $*$ on set $A$ .

If I'm asked to find if there exists a neutral/identity element of a certain table where $a * a, a * b, b * a \text{ and } b * b$
are defined, how do I figure that out?

What I mean is suppose 1 of 16 table of operations looks like this:

$\begin{array}{|l||r|} \hline (a * a) & a \\ \hline (a * b) & b \\ \hline (b * a) & a \\ \hline (b * b) & b \\ \hline \end{array}$

How do I figure out if an identity element exist for this table? Also is it possible to find the identity/neutral element from this table?

**HallsofIvy** · August 24th 2012, 05:48 PM

That binary operation does not have an identity. The definition of identity is "e is an identity for operation "*" if and only if e*x= x*e= x for every x in in the set". Here, a*b= b but $b*a= a\ne b$ so a is NOT an identity. Similarly, b*a= a but $a*b= b\ne a$ so b is not an identity.

**x3bnm** · August 24th 2012, 07:15 PM

Again thanks.

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