Spring 2010: MATH 3210-003
Foundations of Analysis I
Wed. 3rd Feb.
Midterm Examination
Student ID:
Name:
Instructions.
1. Attempt all questions.
2. Scratch paper is on the last page.
Question Points Your Score
Q1
5
Q2
5
Q3
5
Q4
5
TOTAL
20
Dr. R. Lodh
50 minutes
Q1]. . . [5 points] Let f : X → Y be a function and A ⊂ X a subset. Prove that if f is one-to-one then
f (X \ A) = f (X) \ f (A).
OVER
Q2]. . . [5 points] Let a ≥ 0 be a real number. Prove that
(1 + a)n ≥ 1 + na
for any n ∈ N.
Q3]. . . [5 points] Recall that a commutative ring R is a set with two binary operations (+, ×) satisfying
the following axioms (where we write xy for x × y):
A1 x + y = y + x for all x, y ∈ R
A2 x + (y + z) = (x + y) + z for all x, y, z ∈ R
A3 there is 0 ∈ R such that x + 0 = x for all x ∈ R
A4 for each x ∈ R there is −x ∈ R such that x + (−x) = 0
M1 xy = yx for all x, y ∈ R
M2 x(yz) = (xy)z for all x, y, z ∈ R
M3 there is 1 ∈ R, 1 6= 0, such that 1x = x for all x ∈ R
D x(y + z) = xy + xz for all x, y, z ∈ R.
Using only the above axioms, prove that 0x = 0 for all x ∈ R. In your proof mention each axiom whenever
you use it.
Q4]. . . [5 points] Let f : X → R be a function and c ∈ R. Prove that for any subset A ⊂ X we have
sup(cf ) = c sup(f ).
A
A
Scratch Paper