In exercise 11.2.3 : [ The exercise is below the question items]

i. Items (b), (d), (e), and (f) are groups. For each, show what the identity is, and what the inverse of atypical element is.

ii. Item (a) is not a group. Why not?

iii. Item (g) is not a group. Why not?

iv. Item (c) does not even specify an operator. Why not

EXERCISE: 11.2.3 :( This is the Exercise that the above question is referencing)

3. Which of the following are groups? ( do not answer this; this is the reference for the above question)

(a) B with concatenation (see Subsection 11.2.1).

(b) M₂₃(R) with matrix addition. (c) M₂₃(R) with matrix multiplication.

(d) The positive real numbers, R+,with multiplication.

(e) The nonzero real numbers, R, with multiplication.

(f) {1,1} with multiplication.

(g) The positive integers with the operation M defined by aMb = the larger of a and b.