handout
... (Review from Calculus III -- see Stewart Chapter 12) The following homework exercises are due Thursday 12-2-10. A. Sequences. Definition. A sequence in R is a function from the natural numbers {1, 2, 3, ...} to R. We usually write the function values with subscripts instead of function notation. For ...
... (Review from Calculus III -- see Stewart Chapter 12) The following homework exercises are due Thursday 12-2-10. A. Sequences. Definition. A sequence in R is a function from the natural numbers {1, 2, 3, ...} to R. We usually write the function values with subscripts instead of function notation. For ...
Monday, August 8: Samples of Proofs
... Let a b = a + b + 4. Then a k = a + k + 4 and if a + k + 4 = a, then k = -4. Consider k = 4. a 4 = a + 4 + 4 = a for all real values of a, and 4 a = 4 + a + 4 = a for all real numbers a. Hence if a b = a + b + 4 , then a 4 = 4 a = a for all a Reals. Proof by Induction: (Set up ...
... Let a b = a + b + 4. Then a k = a + k + 4 and if a + k + 4 = a, then k = -4. Consider k = 4. a 4 = a + 4 + 4 = a for all real values of a, and 4 a = 4 + a + 4 = a for all real numbers a. Hence if a b = a + b + 4 , then a 4 = 4 a = a for all a Reals. Proof by Induction: (Set up ...
3.1 Functions A relation is a set of ordered pairs
... When we use a formula to define a function we often give a name to a function(f,g,h,etc.) and use a special notation for the output, y. If x is the input, then we denote the output by f(x) (read “f of x”). Caution: f(x) is not a multiplication of f and x. It is an entity that can’t be split. f(x) de ...
... When we use a formula to define a function we often give a name to a function(f,g,h,etc.) and use a special notation for the output, y. If x is the input, then we denote the output by f(x) (read “f of x”). Caution: f(x) is not a multiplication of f and x. It is an entity that can’t be split. f(x) de ...
Math 3000 Section 003 Intro to Abstract Math Homework 8
... 8. Exercise 9.60: Let f : A → B be a function. For a subset C of A, the image of C under f is the set f (C) = {f (c) : c ∈ C}. (Thus f (A) is the range of f .) Let A1 , A2 ⊆ A. Prove the following. (a) f (A1 ∪ A2 ) = f (A1 ) ∪ f (A2 ) (b) f (A1 ∩ A2 ) ⊆ f (A1 ) ∩ f (A2 ) (c) If f is one-to-one, then ...
... 8. Exercise 9.60: Let f : A → B be a function. For a subset C of A, the image of C under f is the set f (C) = {f (c) : c ∈ C}. (Thus f (A) is the range of f .) Let A1 , A2 ⊆ A. Prove the following. (a) f (A1 ∪ A2 ) = f (A1 ) ∪ f (A2 ) (b) f (A1 ∩ A2 ) ⊆ f (A1 ) ∩ f (A2 ) (c) If f is one-to-one, then ...
simultaneous convergence of two sequences
... α + β < 1 and let (xn )n≥1 and (yn )n≥1 be two sequences such that yn = xn + α sin xn−1 + β arctan xn−2 , for all n ∈ N, n ≥ 3. Then the sequence (xn )n≥1 is convergent if and only if the sequence (yn )n≥1 is convergent. Moreover, if the sequence (xn )n≥1 converges to x∞ , then the sequence (yn )n≥1 ...
... α + β < 1 and let (xn )n≥1 and (yn )n≥1 be two sequences such that yn = xn + α sin xn−1 + β arctan xn−2 , for all n ∈ N, n ≥ 3. Then the sequence (xn )n≥1 is convergent if and only if the sequence (yn )n≥1 is convergent. Moreover, if the sequence (xn )n≥1 converges to x∞ , then the sequence (yn )n≥1 ...
