Chapter 14.5, Problem 1P

Consider a function f : S→ T defined by f(n) = n², where the domain is the set of consecutive integers S = {-7,..., -2, -1, 0, 1, 2, ..., 7} and the range is the set of consecutive squares T = {-64,...,-9, -4,-1,0, 1, 4, 9, ..., 64}. Which one of the following choices correct describes f? O one-to-one, but NOT onto Oonto, but NOT one-to-one O one-to-one and onto O neither one-to-one nor onto O none of these

Explain the Wronskian determinant test. Using the Wronskian determinant test, write the program using NumPy to determine whether the functions f(x)=e^(- 3x), g(x)=cos2x and h(x)=sin2x are linearly independent in the range (-∞, + ∞). #UsePython

Let f (n) and g(n) be functions with domain {1, 2, 3, . . .}. Prove the following: If f(n) = O(g(n)), then g(n) = Ω(f(n)).

Simplify the following function: F (W, X, Y, Z) = ∑ (3, 5, 11, 15) with don't care conditions d = ∑(1, 2, 6, 9) . Make a four-variable k-map to simplify F.

Q. Let A = {a, b, c, d, e} and B = {1, 2, 3, 4, 5, 6, 7, 8}. How many functions f : A → B(a)  ... are injective?(b) ... are not injective?(c) ... are such that f(a) = f(b) = f(c)?(d) ... are such that exactly three elements of A have 8 as an image?(e)  ... are surjective?

Program the Gaussian elimination method with no partial pivoting for solving a linear system of the form Ax=b, where b is a single column vector. Your function should take in a coefficient matrix A, and a single input vector b. Your function should return the solution vector x. Your code should also return the appropriate error message. The first line of your function should look like: function x = gaussElimination (A,b)

Reduce the following functions using Karnaugh Map method. F(A, B, C, D) = t (4, 5, 6, 7, 8, 12, 13). d (1, 15).

Determine whether each of the following functions f : {a,b,c,d} -> {a,b,c,d} is one-to-one and/or onto.  (a) f(a) = b, f(b) = a, f(c) = b, f(d) = c (b)  f(a) = b, f(b) = b, f(c) = d, f(d) = c (c)  f(a) = b, f(b) = a, f(c) = c, f(d) = d (d)  f(a) = d, f(b) = a, f(c) = c, f(d) = b (e)  f(a) = c, f(b) = d, f(c) = a

Let A = {a, b, c, d, e} and B = {1, 2, 3, 4, 5, 6, 7, 8}. How many functions f : A → B(a)  ... are injective?(b)  ... are not injective?(c)... are such that f(a) = f(b) = f(c)?(d)  ... are such that exactly three elements of A have 8 as an image?(e) ... are surjective?