Download MAT137 | Calculus! | Lecture 2 | Second half

MAT137 | Calculus! | Lecture 2 | Second half
Tutorials begin next week.
Problem set 1 is available!
Next class: Limits: 2.1-2.2
Francisco Guevara Parra
L5101
MAT137
18 May 2017
One-to-One Functions
Definition (One-to-One Function)
Let f be a function with domain D.
We say that f is one-to-one if each different input produces a different
output.
g
f
one-to-one
Not one-to-one
Francisco Guevara Parra
L5101
MAT137
18 May 2017
One-to-One Functions
Which of the following are valid ways to write the definition of one-to-one
function?
Definition
Let f be a function with domain D. We say that f is one-to-one if . . .
1
f (x1 ) 6= f (x2 ).
2
∃x1 , x2 ∈ D
3
∀x1 , x2 ∈ D
f (x1 ) 6= f (x2 ).
4
∀x1 , x2 ∈ D
x1 6= x2 ,
f (x1 ) 6= f (x2 ).
5
∀x1 , x2 ∈ D
x1 6= x2
⇒
6
∀x1 , x2 ∈ D
f (x1 ) 6= f (x2 )
⇒
x1 6= x2 .
7
∀x1 , x2 ∈ D
f (x1 ) = f (x2 )
⇒
x1 = x2 .
s.t.
Francisco Guevara Parra
f (x1 ) 6= f (x2 ).
f (x1 ) 6= f (x2 ).
L5101
MAT137
18 May 2017
Sum of odd numbers
Theorem
The sum of two odd numbers is an even number.
Proof 1 (?)
5 is odd.
7 is odd.
5 + |{z}
7 = |{z}
12 .
|{z}
odd
odd
even
Francisco Guevara Parra
L5101
MAT137
18 May 2017
Sum of odd numbers
Theorem
The sum of two odd numbers is an even number.
Proof 1 (?)
5 is odd.
7 is odd.
5 + |{z}
7 = |{z}
12 .
|{z}
odd
odd
even
| This one was a bad proof!
Francisco Guevara Parra
L5101
MAT137
18 May 2017
Sum of odd numbers
Theorem
The sum of two odd numbers is an even number.
Proof 2 (?)
even + even = even
odd + even = odd
even + odd = odd
odd + odd = even
Francisco Guevara Parra
L5101
MAT137
18 May 2017
Sum of odd numbers
Theorem
The sum of two odd numbers is an even number.
Proof 2 (?)
even + even = even
odd + even = odd
even + odd = odd
odd + odd = even
| This one was also a bad proof!
Francisco Guevara Parra
L5101
MAT137
18 May 2017
Definition: Odd
Which of the following are valid ways to write the definition of an odd
number?
Definition
Let x ∈ Z. We say that x is an odd number if . . .
1
x=2k+1.
2
∀k ∈ Z
x = 2k + 1.
3
∃k ∈ Z
s.t. x = 2k + 1.
Francisco Guevara Parra
L5101
MAT137
18 May 2017
Sum of odd numbers (A correct proof)
Theorem
The sum of two odd numbers is an even number.
Proof
Let x = 2n + 1 and y = 2m + 1, where n, m ∈ Z. Then
x + y = 2n + 1 + 2m + 1
Francisco Guevara Parra
L5101
MAT137
18 May 2017
Sum of odd numbers (A correct proof)
Theorem
The sum of two odd numbers is an even number.
Proof
Let x = 2n + 1 and y = 2m + 1, where n, m ∈ Z. Then
x + y = 2n + 1 + 2m + 1 = 2n + 2m + 2
Francisco Guevara Parra
L5101
MAT137
18 May 2017
Sum of odd numbers (A correct proof)
Theorem
The sum of two odd numbers is an even number.
Proof
Let x = 2n + 1 and y = 2m + 1, where n, m ∈ Z. Then
x + y = 2n + 1 + 2m + 1 = 2n + 2m + 2 = 2(n + m + 1)
Francisco Guevara Parra
L5101
MAT137
18 May 2017
Sum of odd numbers (A correct proof)
Theorem
The sum of two odd numbers is an even number.
Proof
Let x = 2n + 1 and y = 2m + 1, where n, m ∈ Z. Then
x + y = 2n + 1 + 2m + 1 = 2n + 2m + 2 = 2(n + m + 1)
Therefore, x + y = 2k, where k = n + m + 1 ∈ Z. Hence x + y is an even
number.
Francisco Guevara Parra
L5101
MAT137
18 May 2017
All real numbers are equal to 1
Theorem (?)
∀x ∈ R, x = 1.
Proof (?)
Let x ∈ R. Then
x 2 − 2x + 1 = 1 − 2x + x 2
(x − 1)2 = (1 − x)2
Taking square root on both side we get
x − 1 = 1 − x.
so,
2x = 2.
Hence, x = 1.
Francisco Guevara Parra
L5101
MAT137
18 May 2017
Axiom of induction
Let S be a set of positive integers. If:
1
1∈S
2
k ∈ S implies k + 1 ∈ S
then, all positive integers belong to S
Intuitively, 1 ∈ S and k ∈ S implies k + 1 ∈ S gives us: 1 ∈ S implies
1 + 1 ∈ S implies 2 + 1 ∈ S implies 3 + 1 ∈ S, and so on.
Francisco Guevara Parra
L5101
MAT137
18 May 2017
Structure of a proof by induction
A proof by mathematical induction goes like this:
Let S be the set of positive integers n for which the property P(n) is true.
Then follow a two step process:
Francisco Guevara Parra
L5101
MAT137
18 May 2017
Structure of a proof by induction
A proof by mathematical induction goes like this:
Let S be the set of positive integers n for which the property P(n) is true.
Then follow a two step process:
1
Basis: show that 1 ∈ S.
Francisco Guevara Parra
L5101
MAT137
18 May 2017
Structure of a proof by induction
A proof by mathematical induction goes like this:
Let S be the set of positive integers n for which the property P(n) is true.
Then follow a two step process:
1
Basis: show that 1 ∈ S.
2
Inductive hypothesis: assume k ∈ S
Francisco Guevara Parra
L5101
MAT137
18 May 2017
Structure of a proof by induction
A proof by mathematical induction goes like this:
Let S be the set of positive integers n for which the property P(n) is true.
Then follow a two step process:
1
Basis: show that 1 ∈ S.
2
Inductive hypothesis: assume k ∈ S
3
Final step: prove that k + 1 ∈ S.
Francisco Guevara Parra
L5101
MAT137
18 May 2017
Structure of a proof by induction
A proof by mathematical induction goes like this:
Let S be the set of positive integers n for which the property P(n) is true.
Then follow a two step process:
1
Basis: show that 1 ∈ S.
2
Inductive hypothesis: assume k ∈ S
3
Final step: prove that k + 1 ∈ S.
Example: prove that for all positive integer n, 1/n < 2.
Francisco Guevara Parra
L5101
MAT137
18 May 2017