Functions

Let P be the power set of {a,b,c}. A function: f: P -> Z follows: For A in P, f(A) = the number of elements in A. Is f one-to-one? Prove or disprove. Is f onto? Prove or disprove.

Functions

Which functions are one-to-one? Which functions are onto? Describe the inverse function A)F:Z^2-N where f is f(x,y) x^2 +2y^2 B)F:N->N where f is f(x) = x/2 (x even) x+1 (x odd) C)F:N->N where f is f(x) = x+1 (x even) x-1 (x odd) D)h:N^3 -> N where h(x,y,z) = x + y -z

Functions

Find the composition of the following cycles representing permutations on A = {1,2,3,4,5,6,7,8} Answer as a composition of one or more disjoint cycles. A) (1,3,4) . (5,1,2) B) (2,7,8) . (1,2,4,6,8) C) (1,3,4) . (5,6) . (2,3,5) . (6,1)

Recursive definition

I need to give a recursive definition with initial condition(s). a.) The sequence {an}, n = 1,2,3,… where an = 2n. b.) The Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, ….

Mathematical induction

a.) Use the Principle of Mathematical Induction to prove that n3 > n2 + 3 for all n ≥ 2. b.) Use mathematical induction to prove that every amount of postage of six cents or more can be formed using 3-cent and 4-cent stamps.

Matrices (Matrix Product Rule); Geometric Series

1. Suppose that A = ... and C = ... (see attachment). Find a matrix for B such that AB = C or prove that no such matrix exists 2. Find the sum ... (see attachment)

Factoring

a. Convert (11101)2 to base 16. b. Use the Euclidean algorithm to find gcd(34,21).

Big-Oh Function

Please see the attached file for the fully formatted problems. I need to find the best big-oh function for the function. I need to choose my answer from among the following: 1, log2 n, n, n log2 n, n2, n3,…, 2n, n!. A. f(n) = 1 + 4 + 7 + ... + (3n + 1). B.

Big-Oh

Use the definition of the big-oh to prove that ... (see attachment)

Graphs and Their Representations

If all the nodes of a simple, connected, planar graph have degree 4 and the number of arcs is 12, into how many regions does it divide the plane?