Functions and Relations

Question 1

If A and B are sets, a relation from A to B is …

Answer 1

… a subset of the Cartesian product AxB

Question 2

So, a relation R from A to B (sets) is a subset of AxB. Given an ordered pair in AxB (x, y), x is related to y by R, if and only if …

Answer 2

(x, y) is in R

Question 3

Assume a relation R from A to B. What is the domain and codomain of R?

Answer 3

A is the domain and B is the codomain

Question 4

A function F from a set A to a set B is a relation with domain A and codomain B that satisfies the following properties:
…

Answer 4

1. For every element x in A, there is an element y in B, such that (x, y) is in F
2. For all elements x in A and y, z in B,
   if (x, y) is in F and (x, z) is in F, then y = z

Easily put as:

1. Every element of A is the first element of an ordered pair of F
2. No two distinct ordered pairs in F have the same first element

Question 5

Another definition of domain and codomain of a function that I think it is easiest to understand:

Answer 5

The domain is the set in which the variable x takes values.

The codomain is the set in which the function f(x) takes values (the set of the images of the function).

Question 6

Let’s repeat: A function f from a set X to a set Y is a relation from X (the domain) to Y (the codomain) that satisfies two properties:
…

Answer 6

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

SOOOO Every element in the domain is related to one and only one element in Y.

Question 7

What is the range of a function f?

Answer 7

The range is the set of all values of f taken together.

Question 8

The image of X under f = the range of f. True or false?

Answer 8

True

Question 9

What is the set of the images of f?

Answer 9

The set of the images of f is the set of all f values taken together.

Question 10

What is the inverse image of y? (when y = f(x))

Answer 10

The inverse image of y is the set of all values in X that have y as their image.

Question 11

When are two functions equal?

Answer 11

Two functions are equal if they have the same domain and codomain and their values are the same for all elements of the domain.

Question 12

What is a one-to-one function? (injective)

Answer 12

An injective function is a function that maps distinct elements of its domain to distinct elements of its codomain.

Question 13

How do you check whether a function is injective?

Answer 13

you set f(x1) = f(x2) and see if it leads to x1 = x2. If yes, then the function is injective

btw, all linear functions are injective, when a in ax+b=0 is != 0

Question 14

When is a function onto? (surjective)

Answer 14

A function f from a set X to a set Y is surjective, if for every element y in the codomain Y of f, there is at least one element x in the domain of X of f such that f(x) = y

SO, every element of Y has to be related to an least one element of X for the function to be surjective

Question 15

When is a function bijective? (one-to-one correspondence)

Answer 15

When it is both injective and surjective

Question 16

How do you simply define the inverse of a relation?

Answer 16

If R is a relation from A to B, then a relation Rˆ(-1) from B to A can be defined by interchanging the elements of all ordered pairs of R.

Question 17

There are three main properties of relations. Can you name them?

Answer 17

Reflexivity, symmetry, transitivity.

Question 18

When is a relation reflexive?

Answer 18

A relation is reflexive when each element is related to itself.

Question 19

When is a relation symmetric?

Answer 19

A relation is symmetric when, if one element is related to another, then the second element is related to the first.

Question 20

When is a relation transitive?

Answer 20

A relation is transitive when, if one element is related to another, and that latter is related to a third, then the first element is related to that third element.

Question 21

Good to know.

A negation of a universal statement is a ______ statement

Answer 21

Existential

Question 22

The definitions of reflexivity, symmetry and transitivity are all examples of ____ statements

Answer 22

Universal

Question 23

When analysing transitivity, if the hypotheses are not met (if one element is related to another and that other is related to a third) then the relation is __________

Answer 23

Vacuously transitive

Question 24

When is a relation called “equivalence relation”?

Answer 24

When the relation is reflexive, symmetric and transitive

Question 25

If F is a one-to-one correspondence from a set X to a set Y, then there is a function from Y to X that “undoes” the action of F (i.e., sends each element of Y back to the element of X that it came from). This function is called …

Answer 25

… the inverse function for F.

Question 26

(g ○ f) (x) = ?
when we set the property that the range of f is a subset of the domain of g

And, how is this function called?

Answer 26

(g ○ f) (x) = g(f(x))

The function is called the composition of f and g