Question 1:
Let A and B be subsets of the universal set E.
(a) Using set identities (in particular, using only equalities) simplify the following expressions
(i) A \ (A \ B),
(ii) (A ∩ B) \ (A ∩ Bc).
(b) Prove by double inclusion the set identity
A ∩ (A ∪ B) = A.
Question 2:
(a) Let P denote the set of positive numbers, which is assumed to exist. Reformulate the order axioms of R in terms of P.
(b) Show that if 1 < a, then 1 < a < a2
Question 3:
(a) Determine according to the values of the real number x when the rational function
f(x) =( x - 3)/(6x2 - x - 2) is positive, 0, or negative.
(b) Sketch the graph of f : R → R: x → |2x - 1| - |3x + 5| = f(x).
For the sketch take x between -7 and 2, clearly indicating the x- and y-intercepts.
(c) Determine and sketch the set of pairs (x, y) in R×R that satisfy |x y| < 5.
(Hint: One may evaluate |x y| according to the sign of x y. This gives rise to four cases. By letting y be a function of x, the four cases correspond to two graphs. Clearly plot these graphs for -10 ≤ x ≤ 10, indicating only the relevant values for x. Then shade the required area and describe it explicitly.)