Math 117 Fall 2014 Lecture 11
(Sept. 19, 2014)
Reading: 314 Notes: Sets and Functions 3.3, 3.5. Dr. Bowman's book: 3.A, 3.B.
Recall function.
The key words here are to every and exactly one. These are the two things we need
to check when checking whether
Math 117 Fall 2014 Lecture 22
(Oct. 10, 2014)
Reading: Bowman 3.D; 314 Limit & Continuity 3.
Function limit and sequence limit.
Theorem 1. Let f (x): R 7! R and a; L 2 R. Then limx!af (x) = L if and only if for every
sequence fxn g satisfying limn!1xn = a
Math 117 Fall 2014 Homework 3 Solutions
Due Thursday Oct. 2 3pm in Assignment Box
Question 1. (5 pts) Let B Y and f : X 7! Y. Prove that f (f 1(B) B. Can we replace by
=? Justify your claim.
Proof.
f (f 1(B) B. Take an arbitrary y 2 f (f 1(B). By denition
Math 117 Fall 2014 Lecture 14
(Sept. 25, 2014)
Reading: 314 Proof and Logic: 2; 314 Midterm Review A, D.
Mathematical statement:
A statement that is either true or false, but not both.
Propositional Logic: Statements and their combinations, no variable in
Math 117 Fall 2014 Lecture 10
(Sept. 18, 2014)
Reading: 314 Notes: Sets and Functions 3.1, 3.2.
Function.
A function f: X 7! Y is a triplet (X ; Y ; f) where X ; Y are sets and f is a rule assigning
to each element x 2 X exactly one element in Y , this el
Math 117 Fall 2014 Lecture 13
(Sept. 24, 2014)
Reading: Dr. Bowman's book: 1.E, 1.F.
Induction.
To prove that innitely many statements are true. These innitely many statements
must be ordered through a parameter n 2 N.1 That is these statements can be lis
Math 117 Fall 2014 Lecture 16
(Oct. 1, 2014)
Reading:
Recall denition of limit: Let a 2 R, fan g a sequence of real numbers. Say limn!1an = a if
and only if
8" > 0 9N 2 N 8n > N ;
jan aj < ":
(1)
Example 1. Prove limn!12n = 0.
Proof. Let " > 0 be arbitrar
Math 117 Fall 2014 Lecture 15
(Sept. 29, 2014)
Reading: Bowman: 2.A, 2.B.
Sequence.
Definition 1. A sequence of real numbers is a function a: N 7! R.
We often write a(n) as an and denote the whole sequence by fan g or a1; a2; a3; :
Note. The dierence betw
Math 117 Fall 2014 Lecture 17
(Oct. 2, 2014)
Reading: Bowman: 3.C.
In this lecture we discuss limits for a function f : R 7! R.
Denition of limx!af (x) = L for a; L 2 R.
8" > 0 9 > 0 8x0 < jx aj < ;
jf (x) Lj < ":
(1)
Example 1. Prove limx!0x3 = 0 by deni
Math 117 Fall 2014 Lecture 12
(Sept. 22, 2014)
Reading: 314 Notes: Proof and Logic 1.
Steps to approach a (proof) problem.
1. Check the denitions involved.
2. Identify the logical structure:
Is it prove or disprove?
What are the conclusions that need to b