Question

Python

1. Open up IDLE and in a multiline comment write “Functions and Sets II,” your name, and the date.

2. Use Python to write a function f(n) that finds the sum of the first n positive integers. Do not use the sum() function or lists. Write the function from scratch. Then use a for loop to print the sums for n = 10, 50, 1000.

3. Finding the Image of a function Suppose that we have a function f : A → B where A is a finite set. We can use Python to find the image f(A) with a single command, provided we have already defined f as a function. im_f = {f(a) for a in A} For the following functions and domains, use Python to compute and print the images: a) f(x) = x 2 − 2x + 7 and A = {1, 2, 3, . . . , 20}. b) g(x) = (x + 1)2/(3x + 2) and A = {2, 4, 6, 8, 10}.

4. Testing for onto: To determine a function is onto is simple. If f : A → B we first compute im(f) as in the previous exercise. Then we compute any(x for x in B-im_f) which literally asks if there is anything in the codomain that is left when we remove the image. If there are, then the function is not onto. Use Python to determine if f(x) = 2x−1 with domain {1, 2, . . . , 10} is onto the set {1, 3, 5, . . . , 19}.

5. Figure out how to test a function for one to one using the all() command and list comprehension only. No looping, no element-by-element testing allowed! Check your result with the function h(x) = x 3 + 5 over the domain {−100, −99, . . . , 99, 100}

Answer 1

2)

def SUM(n):
    s = 0
    s = (n*(n+1))/2
    print s

#Driver program

a = [100,50,1000]
for x in a:
    SUM(x)

3)

def funcF(A):
    print "Images of function f(x) are :\n"
    for x in A:
        s = x+2-2*x+7
        print "%d --> %d"%(x,s)

def funcG(A):
    print "\n\nImages of function g(x) are : \n"
    for x in A:
        x = float(x)
        s = float(((x+1)*2)/(3*x+2))
        print "%d --> %f"%(x,s)

A = [x for x in range(1,20)]

funcF(A)

A = [x for x in range(2,11,2)]

funcG(A)