Functions

Question 1

In functions

What does f: X → Y mean?

Answer 1

A function f from set X to a set Y. In other words, it is a relation of X (the domain of f) to Y (the co-domain of f)

Question 2

What two properties must a function f: X → Y satisfy?

Answer 2

1. Every element of X is related to some element in Y
2. No element of X is related to more than one element in Y

Question 3

What is meant by “f sends x to y”

Answer 3

For every element x, in set X, there is a unique element y, in set Y that is related to x by f.

Question 4

True or false?

The following defines a function. Explain why or why not

Answer 4

True

Question 5

Answer 5

Only (c) defines a function. In (a), not every element of X is related to Y. In (b) there is an element in X such that X is related to more than one elements of Y (disobeys properties of functions).

Think of it like graph functions. y=0 can have muliple x points (parabola, cubic) but each x value must have only one y value related to it.

Question 6

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Question 7

Answer 7

f(a) = 2, f(b) = 4, f(c) = 2

Question 8

Answer 8

range of f = {2, 4} as these are the elements of Y that are related to X

Question 9

Answer 9

c is indeed an inverse image of 2 as p(c)=2. but b is not an inverse image of 3 because p(b) ≠ 3

Question 10

Answer 10

Question 11

Answer 11

{(a, 2), (b, 4), (c, 2)}

Question 12

True or false?

There can be elements of the co domain to which no relation is made from the domain.

Answer 12

True. This means that given a function relating X to Y. There can be elements in Y such that X does not relate to Y

Question 13

Given F: X → Y and G: X → Y

When is are the two functions F and G equal (F = G)

Answer 13

F = G if and only if F(x) = G(x) for every x in the set of X

Question 14

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Question 15

Answer 15

Remember to treat F+G and G+F as functions

Question 16

Given a set X and I_X: X → X

What does I_X(x) = x define

X = capital X
x = lowercase x

Answer 16

The identity function on X because it send each element of X to the element that is identical to it.

Thus the identity function can be pictured as a machine that sends each piece of input directly to the output chite without chaninging it in any way.

Question 17

Answer 17

Don’t be confused by all the random symbols

Question 18

Just some revision

Question 19

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Question 20

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Question 21

How do you define (n-place) Boolean function, f

Answer 21

An (n-place) Boolean function, f is a function whose domain is the set of all ordered n-tuples of 0’s and 1’s and co-domain being a set of only {0, 1}

Question 22

Answer 22

Question 23

When is a ‘function’, not well defined

Answer 23

When it fails to satisfy at least one of the requirements for being a function. That is:
For it to not be well defined:
1. an element in the domain has no relationship to any element in the co-domain
2. an element of the domain has a relationship to multiple elements of the co-domain

Question 24

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Question 26

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Question 27

What does an inverse function do?

Answer 27

"undoes" the action of the function. That is, converting the co-domain back to the domain.

Question 28

What is it meant for a function to be one-to-one?

Answer 28

All elements of the domain have only one relation to the co-domain

Question 29

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Question 31

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Question 32

What is an 'onto' or 'surjective' function

Answer 32

Every element of co-domain is an image of an element in the domain. (every element of co-domain is related to an element of the domain) ## Footnote In terms of arrow diagrams, this means a function is onto if each element of the co-domain has an arrow pointing to it from some element of the domain.

Question 33

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Question 37

# Given f: X → Y where X and Y are sets. What is the relationship of X and Y if the function `f` is one-to-one and onto.

Answer 37

1. Each element of X is related to only one element of Y 2. Each element of Y is an image of only one element of X

Question 38

What defines a function being one-to-one correspondence or bijection?

Answer 38

The function is both one-to-one and onto ## Footnote Given f: X → Y for set X and Y 1. One to one: each element of X only has one relation (or arrow) to Y 2. Onto: every element of Y is related (has an arrow) to an element in X

Question 39

Answer 39

Question 40

# Given F: X → Y is a one-to-one correspondence function Since the function is a one-to-one correspondence function, what other function relating X and Y do you know exists?

Answer 40

The inverse function. If you know that F: X → Y is a one-to-one correspondence function, then you must know that there is also a function such that F^-1: Y → X exists

Question 41

Answer 41

Question 42

# Given X and Y are sets. True or false? If F: X → Y is one-to-one and onto, then F^-1:Y → X is also one-to-one and onto.

Answer 42

True.

Question 43

What does g ∘ f mean?

Answer 43

The composition function of **f and g**. Meaning g(f(X)) | Apply function g first then f ## Footnote It needs to be read "compositon of _f and g_" and not "composition of _g and f_" because you apply f first.

Question 44

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Question 45

# True or false? In a composition of an identity function and any function, you can just ignore the identity function. *(Composition function of any function and identity function equals the function)*

Answer 45

True.

Question 46

# Given f: X → Y What is the special function you get when you compose f^-1 and f

Answer 46

The identity function. Specifically one that acts on the set X | I_Y(n) = f^-1

Question 47

# Given f: X → Y What is the special function you get when you compose f and f^-1

Answer 47

The identity function. Specifically one that acts on the set X | I_X(n) = f ∘ f^-1

Question 48

Denote "the function composing of f and g" with the ∘ symbol

Answer 48

g ∘ f | You apply f first. g(f(x))

Question 49

Denote "the function composing of g and f" with the ∘ symbol

Answer 49

f ∘ g | You apply g first. f(g(x))

Question 50

# True or false? The composition function of two one-to-one functions is a one-to-one function.

Answer 50

True

Question 51

# True or false? The composition function of two onto functions is a onto function.

Answer 51

True

Question 52

Prove this statement

Answer 52

Question 53

Prove this statement.

Answer 53

Question 54

What is the term 'cardinal number'

Answer 54

It refers to the size of a set | "This set has *eight* elements"

Question 55

What does the term "ordinal number" refer to?

Answer 55

The order of an element in a sequence | "This is the *eighth* element in the row"

Question 56

# Given and two sets A and B? How do you define if A and B have the same cardinality. | Cardinality refers to the size of the sets.

Answer 56

A has the same cardinality of B if and only if there one-to-one correspondence from A to B. ## Footnote In other words, A has the same cardinality as B if and only if there exists a function from A to B that is both one-to-one and onto.

Question 57

# Given a set A What is the reflexive property of cardinality | Cardinality refers to the size of the sets.

Answer 57

A has the same cardinality as A | Just remember reflexive means it's common sense; a reflexive instinct

Question 58

# Given sets A and B What is the symmetric property of cardinality | Cardinality refers to the size of the sets

Answer 58

If A has the same cardinality as B then B has the same cardinality of A. ## Footnote Therefore, you can say that A and B have the same cardinality

Question 59

# Given sets A, B and C What is the transistivity propert of cardinality? | Cardinality refers to the size of the sets

Answer 59

If A has the same cardinality as B and B has the same cardinality as C then A has the same cardinality as C.

Question 60

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Question 61

# GIven a set A When can A be called *countably infinite*

Answer 61

**If A has the same cardinality Z⁺.** The reason is that there would exist a function that "counts" the elements of A if this statement is true. Like every element of A will be linked to a positive integer. This function would be one-to-one correspondence. F: Z⁺ → A ## Footnote Remember that same cardinality means there exists a function that is one-to-one correspondence linking the two sets.

Question 62

Just revision