Lecture 1

Question 1

What are functions?

Answer 1

A process that associates one set (input) with another set (output); forms a relationship between two sets

A rule that maps a unique element of the domain to ONLY ONE other unique element of the codomain.

Question 2

What are sets?

Answer 2

A set is an unordered collection of distinct/unique ‘objects/entities/elements’

NOTE: Cannot have any repeating elemenets, unlike an array/list

Question 3

What are the notations used for sets? What do they define?

Answer 3

x ∈ B if x is an element of B
x ∈/ B if x is not in B

A ⊆ B is every a ∈ A lies also in B,
A is a subset of B

A ∩ B is the set of all elements that lie in A AND in B;
we say A ∩ B is the intersection of A and B.

A ∪ B is the set of ALL elements that lie in A OR in B;
we say A ∪ B is the union of A and B.

A \ B is the set of all elements in A that are NOT in B

Question 4

How do you define by attributes/properties?

Answer 4

If S is a set and, and P is a property of elements in S, then

{x ∈ S : P(x)}

is the set all elements of S which have the property P

e.g. {x ∈ Z : −1 <= x <= 1} = {−1, 0, 1}

Question 5

What are intervals?

Answer 5

[a, b] = {x ∈ R : a <= x <= b}
[] = closed intervals, includes a and b

(a, b) = {x ∈ R : a < x < b}
() = open intervals, excludes a and b

Can mix interval signs
[a, b) = {x ∈ R : a <= x < b}

Question 6

What is Pi notation?

Answer 6

A shorthand notation to write out products in a more compact form.

NOTE: Empty set results in empty product = 1 BY CONVENTION
(techinically undefined)

Question 7

What is Sigma notation?

Answer 7

A way to write sums in a more compact form.

NOTE: Empty set results in empty sum = 0 BY CONVENTION (technically undefined)

If the function is a constant, x acts as a counter only rather than affecting the value.

Question 8

What are images?

Answer 8

The image of f : X → Y is the set f (X) = {f (x) : x ∈ X}.

It is a subset of the codomain that can be reached by elements in the domain using the applied rule; the range of the function.

Question 9

What is Barber’s paradox?

Question 10

What are some important sets?

Answer 10

{} or ∅ is the empty set

N is the set of natural numbers
{0, 1, 2, . . .}

Z is the set of integers
{. . . , −2, −1, 0, 1, 2, . . .}

R is the set of real numbers (all numbers on a
’standard’ number line,
e.g. 5,√2, 2/3, −π, . . .

Question 11

What is Russel’s paradox?

Question 12

True or false? a ∈ {{a, b}, {c, d}, {a}}

Answer 12

False

no element a

only elements {a, b}, {c, d}, {a}

Question 13

List all alements of following sets
1. {x ∈ Z : x ∈ {−1, 0, 1.5}}

2. {x ∈ N : x is odd and x 6 5}

Answer 13

1. {-1, 0}
2. {1, 3, 5}

Answer this by creating a set with the elements, NEEDS {}!!!

Question 14

Can sigma and pi notations have multiple ways to be written down?

Answer 14

Yes

Question 15

What is the domain and codomain?

Answer 15

Domain = where inputs are housed

Codomain = where outputs are housed

Question 16

What does a function need?

Answer 16

• Domain
• Co-domain
• ‘Rule’

Question 17

How do we define function graphs?

Answer 17

The graph (or plot) of f : X → Y is {(x, f (x)) : x ∈ X}

Question 18

What happens if if we don’t specify the codomain in a function?

Answer 18

We choose the image as the codomain.

Question 19

What happens if we don’t specify the domain?

Answer 19

By convention, we choose all real numbers R as the domain X –> R

Question 20

What are zeros or roots?

Answer 20

This is the point of a function where all points x are elements of the domain AND f(x) = 0

Root = x-intercepts!

Question 21

What is an inverse function?

Answer 21

A function that has the domain and codomains swapped AND the function takes the inverse of f(x) back to x. Essentially undoes the function applied on x/y.

g(f(x)) = x where g(x) is the inverse function
AND
f(g(y))= y

g(x) = f^-1(x)

MUST GO BOTH WAYS!

Question 22

Do inverse functions always exist?

Answer 22

NO, but if they do then they are unique!

Question 23

Ways to tell if a function does not have an inverse?

Answer 23

If the domain and co-domain cannot be reversed without any conflicts

If there are no unique choices when finding the unique element to reverse.

Question 24

Ways to tell if a function does not have an inverse?

Answer 24

If the domain and co-domain cannot be reversed without any conflicts

If there are no unique choices when finding the unique element to reverse. REMEMBER, each element needs to have a unique element between and domain and codomain to determine inverse.

f has an inverse function if and only if it is bijective

Question 25

How do we find the inverse?

Answer 25

f(x) = y, solve for x

Question 25

Define injective?

Answer 25

Means that for all elements of X, there is a unique element in Y when the function f(x) is applied to these elements.

Distinct inputs give distinct outputs.

Question 26

Define surjective

Answer 26

If the image of function f is Y, that is Y = f(X). THE IMAGE IS EQUAL TO CODOMAIN!

for every y ∈ 2 there is some x ∈ X with f (x) = y

Question 27

Define bijective?

Answer 27

When a function is BOTH injective and surjective.

Question 28

How to determine if function is injective/surjective/bijective?

Answer 28

For injective:
Determine if f(x1) = f(x2) where x1, x2 ∈ R
if f(x1) = f(x2) true AND x1 =/ x2 , then NOT injective

For surjective:
Take arbritrary number and show that we can reach it as an image.
if y ∈ R, then y = f(x) and solve for x
If real number AND is within domain then surjective

Question 29

How to define a convex function?

Answer 29

No two points are below the function

Question 30

How to define a concave function?

Answer 30

No two points are above the function

Question 31

What is a linear function?

Answer 31

f(x) = mx + b
m is the slope of f
b = f(0) is the y-intercept

Question 32

How to calculate m?

Answer 32

y2-y1 / x2 - x1