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Material Type: Assignment; Class: Fundamental Mathematics; Subject: Mathematics; University: University of Illinois - Urbana-Champaign; Term: Unknown 2006;
Typology: Assignments
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Homework 5 -Math 347 C
Submission in pairs. Due date: Friday February 24.
(1) 3. Practice problems
Quantifiers: True or false (in R)
(1) ∀x ∃y ∀z ∃w : (x < y ∧ 2 zw < 3 yw^2 ) (2) ∀x ∃y ∀z ∃w : (x < y ∧ 2 zw > 3 yw^2 ) (3) ∀x ∃y ∀z ∃w : (x < y ⇒ zxw < yw) (4) ∀x ∃y ∀z ∃w : (x < y ∧ zxw^2 > zyw^2 ) (5) ∀x ∃y ∀z ∃w : (x < y ∧ zxw > zyw)
Estimates:
(1) Let a > 0 and |x| ≤ min(1, ε). Show that |a^3 − (a + x)^3 | ≤ (a + 1)^3 ε. (Hint 3a^2 + 3a + 1 ≤ (a + 1)^3 .) (2) Show that ∀x (|x| ≤ 11 ⇒ | 6 −
36 + x| ≤ | 11 x|). (3) 3.
Induction:
(1) 3. (2) 6.
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