This is Abstract Algebra and requires proving (proof)

1. Let $ be the subset of R x R defined by o = {(x, 2x - 7) : r ER}. Show that o is a one to one and onto function. 2. Determine whether the binary operation * defined is commutative and whether * is associative. Justify your answer. a. * defined on Q by letting a * b = ab - 1. b. * defined on Q by letting a * b = c. * defined on N by letting a * b = a". 3. Determine whether the definition of * give a binary operation on the set. In the event that * is not a binary operation, state whether Condition 1, Condition 2, or booth of these conditions on our lecture (Page 12 of Chap 1 Preliminary Concepts 3) are violated. a. On N, define * by letting a * b = a - b. b. On N, define * by letting a * b = ab. c. On N, define * by letting a * b = c, where c is the smallest integer greater than both a and b. d. On N, define * by letting a * b = c, where c is at least 3 more than a and b. a. On N, define * by letting a*b = c, where c is the largest integer less than the product of a and b. 4. Prove or disprove: a. Every binary operation on a set consisting of a single element is both commutative and associative. b. Every commutative binary operation on a set having just two elements is associative