6 The binary operation ◇ is defined on the set \(\mathbb { C }\) of complex numbers by
\(( a + i b ) \diamond ( c + i d ) = a c + i ( b + a d )\)
where \(a , b , c\) and \(d\) are real numbers.
- Is \(\mathbb { C }\) closed under △ ? Justify your answer.
- Prove that ◇ is associative on \(\mathbb { C }\).
- Determine the identity element of \(\mathbb { C }\) under \(\diamond\).
- Determine the largest subset S of \(\mathbb { C }\) such that \(( \mathrm { S } , \diamond )\) is a group.