Solutions to Homework #1
(W4150 ‘Intro to Probability and Statistics’, S04)
Sec. 2.2. (1)
(a) S = {8,16,24,32,40,48}
(b) x
2
+ 4x – 5 = (x + 5)(x – 1) = 0
x = -5 and x =1, i.e. S = {-5,1}
(c) S = {T, HT, HHT, HHH}
(d) S = {N. America, S. America, Europe, Asia, Africa, Australia, Antartica}
(e) Solving 2x – 4 >
0 gives x >
2; but we must have also x < 1
S =
∅
Sec. 2.2. (4)
(a) S = {(1,1), (1,2), …., (6,5), (6,6)} – set which consists of all possible pairs of (x,y) where
both x and y can assume values of 1,2,3,4,5 and 6. The same could be expressed as a rule:
(b) S = {(x,y)| 1 <
x,y <
6}
Sec. 2.2. (8)
(a) A = {(x,y)| x + y > 8} = {(3,6), (4,5), (4,6), (5,4), (5,5), (5,6), (6,3), (6,4), (6,5), (6,6)}
(b) B = {(x,2)
∪
(2,y)| 1 <
x,y <
6} = {(1,2), (2,2), (3,2), (4,2), (5,2), (6,2), (2,1), (2,2), (2,3),
(2,4), (2,5), (2,6)}
(c) C = {(x,y)| 5 <
x <
6, 1 <
y <
6} = {(5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3),
(6,4), (6,5), (6,6)}
(d) A
∩
C = {(5,4), (5,5), (5,6), (6,3), (6,4), (6,5), (6,6)}
(e) A
∩
B =
∅
(f) B
∩
C = {(5,2), (6,2)}
(g)
Sec. 2.2. (16)
(a) M
∪
N = {x | 0 < x < 9}
(b) M
∩
N = {x | 1 < x < 5}
(c) M’
∩
N’ = {x | 9 < x < 12}