MAT 137 LEC 0601 Instructor: Alessandro Malus TA: Muhammad Mohid - - PowerPoint PPT Presentation

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Oct 10, 2023 234 likes •549 views

MAT 137 LEC 0601 Instructor: Alessandro Malus TA: Muhammad Mohid September 18th, 2020 Warm-up question : Is this a good definition? Definition We say that x is brutal if a < x 2 , a is even. Why all this fuss about quantifiers

SLIDE 1

MAT 137 — LEC 0601

Instructor: Alessandro Malusà TA: Muhammad Mohid September 18th, 2020 Warm-up question: Is this a good definition?

Definition

We say that x is brutal if a < −x2, a is even.

SLIDE 2

Why all this fuss about quantifiers anyways?

The cake was delicious.

SLIDE 3

Why all this fuss about quantifiers anyways?

The cake was delicious. The cake I baked was delicious.

SLIDE 4

Why all this fuss about quantifiers anyways?

The cake was delicious. The cake I baked was delicious. I baked a cake. The cake was delicious.

SLIDE 5

What is wrong with this proof?

Theorem

The sum of two odd numbers is even.

Proof

5 is odd. 3 is odd. 5 + 3 = 8 is even.

SLIDE 6

What is wrong with this proof? (2)

Theorem

The sum of two odd numbers is even.

SLIDE 7

What is wrong with this proof? (2)

Theorem

The sum of two odd numbers is even.

Proof.

The sum of two odd numbers is always even. even + even = even even + odd = odd

dd + even = odd
dd + odd = even.

SLIDE 8

Definition of odd and even

Write a definition of “odd integer" and “even integer".

SLIDE 9

Definition of odd and even

Write a definition of “odd integer" and “even integer".

Definition

Let x ∈ Z. We say that x is odd when ...

1 x = 2a + 1 ? 2 ∀a ∈ Z, x = 2a + 1? 3 ∃a ∈ Z s.t. x = 2a + 1?

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What is wrong with this proof? (3)

Theorem

The sum of two odd numbers is always even.

SLIDE 11

What is wrong with this proof? (3)

Theorem

The sum of two odd numbers is always even.

Proof.

x = 2a + 1 odd y = 2b + 1 odd x + y = 2n even 2a + 1 + 2b + 1 = 2n 2a + 2b + 2 = 2n a + b + 1 = n

SLIDE 12

Write a correct proof!

Theorem

The sum of two odd numbers is always even.

SLIDE 13

One-to-one functions

Let f be a function with domain D. f is one-to-one means that ...

... different inputs (x) ...
... must produce different outputs (f (x)).

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One-to-one functions

Let f be a function with domain D. f is one-to-one means that ...

... different inputs (x) ...
... must produce different outputs (f (x)).

Write a formal definition of “one-to-one".

SLIDE 15

One-to-one functions

Definition: Let f be a function with domain D. f is one-to-one means ...

1 f (x1) = f (x2) 2 ∃x1, x2 ∈ D, f (x1) = f (x2) 3 ∀x1, x2 ∈ D, f (x1) = f (x2) 4 ∀x1, x2 ∈ D, x1 = x2, f (x1) = f (x2) 5 ∀x1, x2 ∈ D, x1 = x2 =

⇒ f (x1) = f (x2)

6 ∀x1, x2 ∈ D, f (x1) = f (x2) =

⇒ x1 = x2

7 ∀x1, x2 ∈ D, f (x1) = f (x2) =

⇒ x1 = x2

SLIDE 16

One-to-one functions

Let f be a function with domain D. What does each of the following mean?

1 f (x1) = f (x2) 2 ∃x1, x2 ∈ D, f (x1) = f (x2) 3 ∀x1, x2 ∈ D, f (x1) = f (x2) 4 ∀x1, x2 ∈ D, x1 = x2, f (x1) = f (x2) 5 ∀x1, x2 ∈ D, x1 = x2 =

⇒ f (x1) = f (x2)

6 ∀x1, x2 ∈ D, f (x1) = f (x2) =

⇒ x1 = x2

7 ∀x1, x2 ∈ D, f (x1) = f (x2) =

⇒ x1 = x2

SLIDE 17

Proving a function is one-to-one

Definition

Let f be a function with domain D. We say f is one-to-one when

∀x1, x2 ∈ D, x1 = x2 =

⇒ f (x1) = f (x2)

OR, equivalently,

∀x1, x2 ∈ D, f (x1) = f (x2) = ⇒ x1 = x2

SLIDE 18

Proving a function is one-to-one

Definition

Let f be a function with domain D. We say f is one-to-one when

∀x1, x2 ∈ D, x1 = x2 =

⇒ f (x1) = f (x2)

OR, equivalently,

∀x1, x2 ∈ D, f (x1) = f (x2) = ⇒ x1 = x2 Suppose I give you a specific function f and I ask you to prove it is one-to-one.

SLIDE 19

Proving a function is one-to-one

Definition

Let f be a function with domain D. We say f is one-to-one when

∀x1, x2 ∈ D, x1 = x2 =

⇒ f (x1) = f (x2)

OR, equivalently,

∀x1, x2 ∈ D, f (x1) = f (x2) = ⇒ x1 = x2 Suppose I give you a specific function f and I ask you to prove it is one-to-one.

Write the structure of your proof (how do you begin? what

do you assume? what do you conclude?) if you use the first definition.

Write the structure of your proof if you use the second

definition.

SLIDE 20

Proving a function is NOT one-to-one

Definition

Let f be a function with domain D. We say f is one-to-one when

∀x1, x2 ∈ D, x1 = x2 =

⇒ f (x1) = f (x2)

OR, equivalently,

∀x1, x2 ∈ D, f (x1) = f (x2) = ⇒ x1 = x2

SLIDE 21

Proving a function is NOT one-to-one

Definition

Let f be a function with domain D. We say f is one-to-one when

∀x1, x2 ∈ D, x1 = x2 =

⇒ f (x1) = f (x2)

OR, equivalently,

∀x1, x2 ∈ D, f (x1) = f (x2) = ⇒ x1 = x2 Suppose I give you a specific function f and I ask you to prove it is not one-to-one.

SLIDE 22

Proving a function is NOT one-to-one

Definition

Let f be a function with domain D. We say f is one-to-one when

∀x1, x2 ∈ D, x1 = x2 =

⇒ f (x1) = f (x2)

OR, equivalently,

∀x1, x2 ∈ D, f (x1) = f (x2) = ⇒ x1 = x2 Suppose I give you a specific function f and I ask you to prove it is not one-to-one. You need to prove f satisfies the negation of the definition.

Write the negation of the first definition.
Write the negation of the second definition.
Write the structure of your proof.

SLIDE 23

Proving a theorem

Theorem

Let f be a function with domain D.

IF f is increasing on D
THEN f is one-to-one on D

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Proving a theorem

Theorem

Let f be a function with domain D.

IF f is increasing on D
THEN f is one-to-one on D

1 Remind yourself of the precise definition of “increasing" and

“one-to-one".

SLIDE 25

Proving a theorem

Theorem

Let f be a function with domain D.

IF f is increasing on D
THEN f is one-to-one on D

1 Remind yourself of the precise definition of “increasing" and

“one-to-one".

2 To prove the theorem, what will you assume? what do you want to

show?

SLIDE 26

Proving a theorem

Theorem

Let f be a function with domain D.

IF f is increasing on D
THEN f is one-to-one on D

1 Remind yourself of the precise definition of “increasing" and

“one-to-one".

2 To prove the theorem, what will you assume? what do you want to

show?

3 Look at the part you want to show. Based on the definition, what is

the structure of the proof?

SLIDE 27

Proving a theorem

Theorem

Let f be a function with domain D.

IF f is increasing on D
THEN f is one-to-one on D

1 Remind yourself of the precise definition of “increasing" and

“one-to-one".

2 To prove the theorem, what will you assume? what do you want to

show?

3 Look at the part you want to show. Based on the definition, what is

the structure of the proof?

4 Complete the proof.

SLIDE 28

DISproving a theorem

FALSE Theorem

Let f be a function with domain D.

IF f is one-to-one on D
THEN f is increasing on D

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DISproving a theorem

FALSE Theorem

Let f be a function with domain D.

IF f is one-to-one on D
THEN f is increasing on D

1 This theorem is false. What do you need to do to prove it is false?

SLIDE 30

DISproving a theorem

FALSE Theorem

Let f be a function with domain D.

IF f is one-to-one on D
THEN f is increasing on D

1 This theorem is false. What do you need to do to prove it is false? 2 Prove the theorem is false.

SLIDE 31

Before next class...

Watch videos 1.14, 1.15.
Download the next class’s slides (no need to look at them!)