4 A binary operation, ○, is defined on a set of numbers, \(A\), in the following way.
\(a \circ b = \mathrm { k } _ { 1 } \mathrm { a } - \mathrm { k } _ { 2 } \mathrm {~b} + \mathrm { k } _ { 3 }\), for \(a , b \in A\),
where \(k _ { 1 } , k _ { 2 }\) and \(k _ { 3 }\) are constants (which are not necessarily in \(A\) ) and the operations addition, subtraction and multiplication of numbers have their usual notation and meaning. You are initially given the following information about ○ and \(A\).

• \(A = \mathbb { R }\)
• \(0 \circ 0 = 2\)
• An identity element, \(e\), exists for ∘ in \(A\)
  1. Show that \(a \circ b = a + b + 2\).
  2. State the value of \(e\).
  3. Explain whether ○ is commutative over \(A\).
  4. Determine whether or not ( \(A , \circ\) ) is a group.
  5. Explain whether your answer to part (d) would change in each of the following cases, giving details of any change.
     1. \(A = \mathbb { Z }\)
     2. \(A = \{ 2 m : m \in \mathbb { Z } \}\)
     3. \(\quad A = \{ n : n \in \mathbb { Z } , n \geqslant - 2 \}\)