advertisement

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