Question

Let A  \[=\] R  \[\times\] R and \[*\]  be a binary operation on A defined by \[(a, b) * (c, d) = (a + c, b + d) .\] . Show that \[*\] is commutative and associative. Find the binary element for \[*\] on A, if any.

Answer 1

We have,

A \[=\] R \[\times\] R and \[*\] is a binary operation on A defined by \[\left( a, b \right) * \left( c, d \right) = \left( a + c, b + d \right)\]

Now,

 \[\left( a, b \right) * \left( c, d \right) = \left( a + c, b + d \right) = \left( c + a, d + b \right)\] 
\[ \Rightarrow \left( a, b \right) * \left( c, d \right) = \left( c, d \right) * \left( a, b \right)\]

So, \[*\] is commutative.

Also,

\[\left( a, b \right) * \left[ \left( c, d \right) * \left( e, f \right) \right] = \left( a, b \right) * \left( c + e, d + f \right)\] 
\[ = \left( a, b \right) * \left( c + e, d + f \right)\] 
\[ = \left( a + c + e, b + d + f \right)\] 
\[ = \left( a + c, b + d \right) * \left( e, f \right)\] 
\[ = \left[ \left( a, b \right) * \left( c, d \right) \right] * \left( e, f \right)\] 
\[ \Rightarrow \left( a, b \right) * \left[ \left( c, d \right) * \left( e, f \right) \right] = \left[ \left( a, b \right) * \left( c, d \right) \right] * \left( e, f \right)\]

So,  \[*\] is associative .

Let (x, y) be the binary element for \[*\] on A .

\[\left( a, b \right) * \left( x, y \right) = \left( a, b \right) = \left( x, y \right) * \left( a, b \right)\] 
\[ \Rightarrow \left( a + x, b + y \right) = \left( a, b \right)\] 
\[ \Rightarrow a + x = a\text{ and } b + y = b\] 
\[ \Rightarrow x = 0 \text{ and } y = 0\]

Hence, (0, 0) is the binary element for \[*\] on A.