x = range(1,10)
y = range(8,20)

(The answer I am looking for would be the integers 8 and 9.)

>>> x = range(1,10)
>>> y = range(8,20)
>>> xs = set(x)
>>> ys = set(y)
>>> xs.intersection(ys)
set([8, 9])

>>> x = range(1,10)
>>> y = range(8,20)
>>> list(set(x) & set(y))
[8, 9]

range(max(x[0], y[0]), min(x[-1], y[-1])+1)

x = range(1,10)
y = range(8,20)

z = [i for i in x if i in y]
print z

for a,b,c,d in ((1,10,10,14),
                (1,10,9,14),
                (1,10,4,14),
                (1,10,4,10),
                (1,10,4,9),
                (1,10,4,7),
                (1,10,1,7),
                (1,10,-3,7),
                (1,10,-3,2),
                (1,10,-3,1),
                (1,10,-11,-5)):
    x = range(a,b)
    y = range(c,d)
    print 'x==',x
    print 'y==',y
    b = not ((x[-1]<y[0]) or (y[-1]<x[0]))
    print '    x %s y' % ("doesn't overlap","   OVERLAPS  ")[b]
    print

x== [1, 2, 3, 4, 5, 6, 7, 8, 9]
y== [10, 11, 12, 13]
    x doesn't overlap y

x== [1, 2, 3, 4, 5, 6, 7, 8, 9]
y== [9, 10, 11, 12, 13]
    x    OVERLAPS   y

x== [1, 2, 3, 4, 5, 6, 7, 8, 9]
y== [4, 5, 6, 7, 8, 9, 10, 11, 12, 13]
    x    OVERLAPS   y

x== [1, 2, 3, 4, 5, 6, 7, 8, 9]
y== [4, 5, 6, 7, 8, 9]
    x    OVERLAPS   y

x== [1, 2, 3, 4, 5, 6, 7, 8, 9]
y== [4, 5, 6, 7, 8]
    x    OVERLAPS   y

x== [1, 2, 3, 4, 5, 6, 7, 8, 9]
y== [4, 5, 6]
    x    OVERLAPS   y

x== [1, 2, 3, 4, 5, 6, 7, 8, 9]
y== [1, 2, 3, 4, 5, 6]
    x    OVERLAPS   y

x== [1, 2, 3, 4, 5, 6, 7, 8, 9]
y== [-3, -2, -1, 0, 1, 2, 3, 4, 5, 6]
    x    OVERLAPS   y

x== [1, 2, 3, 4, 5, 6, 7, 8, 9]
y== [-3, -2, -1, 0, 1]
    x    OVERLAPS   y

x== [1, 2, 3, 4, 5, 6, 7, 8, 9]
y== [-3, -2, -1, 0]
    x doesn't overlap y

x== [1, 2, 3, 4, 5, 6, 7, 8, 9]
y== [-11, -10, -9, -8, -7, -6]
    x doesn't overlap y

from time import clock

x = range(-12,15)
y = range(-5,3)
te = clock()
for i in xrange(100000):
    w = set(x).intersection(y)
print '                     set(x).intersection(y)',clock()-te


te = clock()
for i in xrange(100000):
    w = range(max(x[0], y[0]), min(x[-1], y[-1])+1)
print 'range(max(x[0], y[0]), min(x[-1], y[-1])+1)',clock()-te

set(x).intersection(y) 0.951059981087
range(max(x[0], y[0]), min(x[-1], y[-1])+1) 0.377761978129