CS 1MD3 - Winter 2006 - Assignment #3

Generated for Student ID: 0570239

Due Date: Monday, February 27th

(You are responsible for verifying that your student number is correct!)
The following exercises involve writing very short (but tricky) functions to manipulate data structures. One of them is a recursive function which may be written in only three lines, while the rest may be written in only one line each (and are not recursive). Function signatures (the "def functionName(x,y,z):" lines) are not counted when enumerating the number of lines. So in this context, a "one-line function" actually occupies two lines: the signature on the first line, and the body on the next line.

Note: Not all students get the same number of questions because some questions are more difficult than others. We have tried to make all the assignments of roughly equal difficulty.

Question 1:

Implement the sumList function (same as sum() for lists) as a recursive function--obviously
without using sum() itself.

Question 2:

A (binary) "symmetric relation," R, is a set of ordered pairs, (x,y), such that 
if (x,y) is in R, then so is (y,x).  For example, the relation
R1 = {(1,2), (3,4), (2,1), (5,5)} is not symmetric, while the relation
R2 = {(1,2), (3,4), (2,1), (5,5), (4,3)} is symmetric.
Write a one-line function which accepts a relation (a set of 2-element tuples), and
returns True if R is symmetric, and False if it is not.

Question 3:

The graph of a (mathematical) function is the set of all pairs (x,y) such that 
x is in the domain of f (i.e. the set of values over which f is defined), and y = f(x).
The range of a function is the set of all values it can attain (i.e. the set of all values
f(x) where x is in the domain of f).
A function is called "injective" (or "one-to-one") if whenever f(x) = f(y) it is necessarily 
the case that x = y.  The function f(x) = 2x is injective, while the function g(x) = x^2 is not
(since g(1) = g(-1) = 1).
Assume f is the graph of a function (no need to specify a domain)