MATH 104-SET THEORY
PROBLEM SET #3
1. Let 𝐴 = {𝑎, 𝑏, 𝑐} and 𝐵 = {1,2}. How many different functions are there from 𝐴 into 𝐵, and
what are they? List all the functions of 𝐴 into 𝐵 by diagrams.
A
1
B
2
C
F- { (A, 1), (A, 2), (B, 1), (B,2) (C, 1) (C, 2)}
2. Let 𝐴 = {𝑎, 𝑏, 𝑐, 𝑑, 𝑒} and let 𝐵 be the set of letters in the alphabet. Let the functions 𝑓, 𝑔, and ℎ of 𝐴
into 𝐵 be defined by:
1) 𝑓(𝑎) = 𝑟, 𝑓(𝑏) = 𝑎, 𝑓(𝑐) = 𝑠, 𝑓(𝑑) = 𝑟, 𝑓(𝑒) = 𝑒
2) 𝑔(𝑎) = 𝑎, 𝑔(𝑏) = 𝑐, 𝑔(𝑐) = 𝑒, 𝑔(𝑑) = 𝑟, 𝑔(𝑒) = 𝑠
3) ℎ(𝑎) = 𝑧, ℎ(𝑏) = 𝑦, ℎ(𝑐) = 𝑥, ℎ(𝑑) = 𝑦, ℎ(𝑒) = 𝑧
State whether or not each of these functions is one-one
1.) Not one-to-one function
f(a)
r
f(b)
a
f(c)
s
f(d)
e
f(e)
2.)
g(a)
A
g(b)
C
g(c)
E
g(d)
R
g(e)
S
One-to-one function
3.) Not one-to-one function.
h(a)
Z
h(b)
Y
h(c)
X
h(d)
h(e)
3. Let 𝐴 = {1,2,3,4,5} and let functions 𝑓: 𝐴 → 𝐴 and 𝑔: 𝐴 → 𝐴 be define by:
𝑓(1) = 3,
𝑓(2) =5,
𝑔(1) = 4,
𝑔(2) = 1,
𝑓(3) = 3,
𝑔(3) = 1,
𝑓(4) = 1,
𝑔(4) = 2,
Find the product functions 𝑓 ∘ 𝑔 and 𝑔 ∘ 𝑓
A
A
f(1)
1
f(2)
2
f(3)
3
f(4)
4
f(5)
5
g(1)
1
g(2)
2
g(3)
3
g(4)
4
g(5)
𝑓(5) = 2
𝑔(5) = 3
4. Let 𝐴 = {1,2,3,4,5}. Let the function 𝑓: 𝐴 → 𝐴 be defined by the diagram
1
1
2
2
3
3
4
4
5
5
Find
𝟏)𝑓 −1 (2) − {2, 4}
𝟐) 𝑓 −1 (3) − {3, ∅}
𝟑) 𝑓 −1 (4) − {(4, 1), (4, 3)}
𝟒) 𝑓 −1 {1,2} − {(1, 2), (2, 4)}
𝟓) 𝑓 −1 {2,3,4} − {(2, 4), (3, ∅), (4, 1), (4, 3)}