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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.
Solution
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.
\[ = \left( a, b \right) * \left( c + e, d + f \right)\]
\[ = \left( a + c + e, b + d + f \right)\]
\[ = \left( a + c, b + d \right) * \left( e, f \right)\]
\[ = \left[ \left( a, b \right) * \left( c, d \right) \right] * \left( e, f \right)\]
\[ \Rightarrow \left( a, b \right) * \left[ \left( c, d \right) * \left( e, f \right) \right] = \left[ \left( a, b \right) * \left( c, d \right) \right] * \left( e, f \right)\]
\[ \Rightarrow \left( a + x, b + y \right) = \left( a, b \right)\]
\[ \Rightarrow a + x = a\text{ and } b + y = b\]
\[ \Rightarrow x = 0 \text{ and } y = 0\]
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