SUNY Geneseo Department of Mathematics
Monday, April 15
Math 239 01
Spring 2019
Prof. Doug Baldwin
Wednesday (April 17).
Lots of math talks and posters; go see them (and other things, too).
Extra credit for writing reactions/reflection on any one math-related presentation.
Dr. Stephanie Singer (also the GREAT Day keynote speaker)
“Defending Democracy with Mathematics”
Thursday, April 18, 2:30 - 3:30
Welles 138
Extra credit for written reactions/reflections, as usual.
An opportunity to learn something about actuarial careers.
Thursday, April 18, 3:45 PM.
South 336.
Prof. (Cesar) Aguilar is looking for 3 summer research assistants to work on graph theory and linear algebra.
A graph in the sense of “graph theory” is a network of “vertices” connected by “edges.” such networks can represent a surprising variety of things, e.g., social media friend relationships (vertices are people, an edge between 2 vertices means that those people are friends).
I strongly recommend research experience for any math student.
Apply by April 22.
See email or Prof. Aguilar for more details.
Section 6.5.
Suppose f : ℝ → ℝ is f(x) = 2x + 1. What is f-1?
Find f-1 by solving f(x) = 2x + 1 (or y = 2x + 1 if you prefer) for x in terms of f(x): f-1(y) = (y-1)/2
How do you know you got it right?
By verifying that f( f-1(x)) = f-1(f(x)) = x
e.g., f( f-1(x)) = f( (x-1)/2 ) = 2 (x-1)/2 + 1 = x - 1 + 1 = x
ℤ5 is the set of integers mod 5, i.e., the set {0,1,2,3,4}.
Write the function g : ℤ5 → ℤ5 defined by g(n) = 3n mod 5 as a set of ordered pairs.
{ (0,0), (1,3), (2,1), (3,4), (4,2) }
Write g-1 as a set of ordered pairs
{ (0,0), (3,1), (1,2), (4,3), (2,4) }
Consider h : ℤ4 → ℤ4 defined by h(n) = 2n mod 4 = { (0,0), (1,2), (2,0), (3,2) }
h-1 = { (0,0), (2,1), (0,2), (2,3) }
This is very much not a function from ℤ4, since it isn’t defined for all elements of ℤ4 and maps the same input to multiple outputs. But it is still the inverse of h.
h-1 is a relation, i.e., a set of ordered pairs from the cross product of a domain and codomain without the restrictions that make some such sets functions.
Definition of inverse, especially as a set of ordered pairs.
Prove that f and g are inverses by showing that f(g(x)) = x = g(f(x))
For f-1 to be a function, f must be a bijection.
See handout for details.
Friday — no class Wednesday.
Introduction to relations.
Read section 7.1