determine and sketch the set of pairs (x,y) in R*R that satisfy the absolute value of x*y < 5

 

 

UNIVERSITY OF VENDA
Department of Mathematics and Applied mathematics MAT 2641 : Real Analysis Assignment 1
Note: • Due Date: By 10:30 AM on Friday 12 August 2016 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 ∩ B c ).
(b) Prove by double inclusion the set identity A ∩ (A ∪ B) = A.
[ 2, 2, 3 Marks]
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 &lt; a, then 1 &lt; a &lt; a2 .
[ 3, 3 Marks]
Question 3:
(a) Determine according to the values of the real number x when the
x−3
is positive, 0, or negative.
rational function f (x) = 2
6x − x − 2
(b) Sketch the graph of f : R → R : x 7→ |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| &lt; 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.)
[ 4, 4, 4 Marks]

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