a) Determine the number of surjective functions ƒ: [4] → [2].
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
100%
Help asap!
![A function f: A → B is surjective if every b E B appears as an actual output, i.e. for
all b € B, there exists some a € A such that f(a) = b. The example function from the
previous question is surjective, whereas g: {1,2,3} → {H,T} given by g(1) = g(2) =
g(3) = T is not surjective.
(a) Determine the number of surjective functions f: [4] → [2].
(b) Suppose an,k is the number of surjective functions f: [n] → [k]. Give a combinato-
rial proof that
n
X
an,k=
k
= x².
k=0
Hint: let k count the number of actual outputs of an appropriate function. You
may find working through the n = 4, x = 2 case enlightening.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8a6e9848-9249-40a2-b72e-f57049e13d57%2Facc6a24c-69fb-47da-9abc-efd0dca2caa3%2Fpewp01_processed.png&w=3840&q=75)
Transcribed Image Text:A function f: A → B is surjective if every b E B appears as an actual output, i.e. for
all b € B, there exists some a € A such that f(a) = b. The example function from the
previous question is surjective, whereas g: {1,2,3} → {H,T} given by g(1) = g(2) =
g(3) = T is not surjective.
(a) Determine the number of surjective functions f: [4] → [2].
(b) Suppose an,k is the number of surjective functions f: [n] → [k]. Give a combinato-
rial proof that
n
X
an,k=
k
= x².
k=0
Hint: let k count the number of actual outputs of an appropriate function. You
may find working through the n = 4, x = 2 case enlightening.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 1 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
