Search & Filtering
Show that A ⊕ B = (A ∪ B) - (A ∩ B).
List the ordered pairs in the relation R from
A = {0, 1, 2, 3, 4} to B = {0, 1, 2, 3}, where (a, b) ∈ R
if and only if
a) a = b. b) a + b = 4.
c) a > b. d) a ∣ b.
e) gcd(a, b) = 1. f ) lcm(a, b) = 2.
For each list of integers, provide a simple formula or rule that generates the terms of an integer sequence that begins with the given list. Assume the sequence starts with . Do not forget initial conditions if required.
a) 7, 11, 15, 19, 23, 27, 31, 35, 39, 43
b) -1, 2, 1, 3, 4, 7, 11, 18, 29, 47
State whether or not the following functions have a well-defined inverse. If the inverse is well-defined, define it. If it is not well-defined, provide justification.
a) f : Z → Z. f(x) = 7x – 7
b) f : R → R. f(x) = 7x – 7
c) A = {a, b, c, d, e}. f : P (A) → {0, 1, 2, 3, 4, 5}. f(x) = |x|. It maps a set to the number of elements it contains.
2. Let S = { :(a1a2a3a4∈ N and 0 ≤ ≤ 9 for each i = 1, 2, 3, 4}. In other words, S is the set of all 4-digit strings with each digit between 0 an 9.
(a) Show that the function f : S → S defined by f(a1a2a3a4) =a4a3a2a1 is a bijection.
(b) (Note that the function reverses the string. For example, f(9527) = 7259) Find f-1 . Specifically, what is f-1.(a1a2a3a4)?
2. Provide the first 5 terms of the following sequence, excluding any initially provided terms. Also, indicate which sequences are increasing, decreasing, non-increasing, and/or non-decreasing.
a) The nth term is ⌈⌉.
b) The nth term is 8.
c) A geometric sequence with the initial term 81 and a ratio of 1/3
2. Indicate if the following statements are possible. Justify your answer. If the answer is “yes”, give a specific example of the functions. Let f: and g: Z be two functions. Use a diagram if that helps explain the function.
a) Is it possible that f is not onto and g ◦ f is onto?
b) Is it possible that g is not onto and g ◦ f is onto?
c) Is it possible that g is not one-to-one and g ◦ f is one-to-one?
2. Consider three functions f, g, and h (from R to R+). Let f (x) = x2, g(x) = 2x and h(x) = log2(x)
a) Evaluate f ◦ g(0)
b) Evaluate h ◦ g(5)
c) Evaluate h ◦ f(0)
d) Give a mathematical expression for h ◦ g
e) Give a mathematical expression for h ◦ g ◦ f
2. For each function below (from Z to Z), indicate whether the function is onto, one-to-one, neither or both. If the function is not onto or not one-to-one, give an example showing why. If the function is bijective, find and show its inverse.
a) f (x) = 3x – 1
b) f(x) = x2 + 2x + 1
c) f(x,y)= 2y -3x
Are the following functions from R to R? If not, explain why
a) f(x) = 1/x
b) f(x) = 2x + 7
