Assignment 5 (More Proofs and Some Functions)

1. A Mathematics Inequality: Use mathematical induction to prove that, given x > 0, (1 + x)ⁿ ≥ 1 + nx for all n ≥ 0.

2. Random Number Generators in C/C++/Python
   The Linux on-line manual states, "The random() function [defined in the <stdlib.h> library] ... returns successive pseudo-random numbers in the range from 0 to (2**31)−1." That is, each time random() is called, it returns a different random number between 0 and 2³¹-1. (More information likely is available from a Linux terminal window by typing man random.)
   Similarly, within Python, the random module, the randint (a, b) function returns a pseudo-random integer N, such that a <= N <= b.
   Pragmatically, it turns out that this function can be extremely useful when programming simulations and other applications.

   1. Within the context of C/C++ and within Python, documentation identifies the random() as a function. With your understanding of a C/C++ or Python, does a random number generater fit within the definition of a "function" for that programming language? That is, what is your understanding of a "function" in a programming language, and does a random number generator satisfy that definition? Explain briefly.

   2. Turning to formal mathematics, does a random number generator, such as random() in C/C++ or randint(a,b) in Python satisfy the definition of a mathematical function? As in Part a, explain briefly.

3. Return Values from Functions in C/C++/Python:
   In C/C++/Python, functions are sometimes described as a block of code, or a set of instructions that perform a specified task. Also, although functions may return a value (with a return statement, this capability is optional.

   Within the framework of formal mathematics, does a function with no return value satisfy the definition of a mathematical function? Explain briefly.

created October 25, 2025 revised October-November, 2025
For more information, please contact Henry M. Walker at walker@cs.grinnell.edu.