2- Functions and Relations

Question 1

What is a relation?

• what does it mean that (x,y) belongs to R? How is it denoted?
• what is the domain of R? the codomain of R?
• make an example of relation and make its notation {(x,y): … with x belongs to X and y belongs to Y}

Answer 1

A relation from set A to set B is a subset of AxB.

• x is related to y by R: xRy
• the domain of R is A;
• the codomain of R is B;

Question 2

What is a function and how is it denoted?
What properties does it need to satisfy?

Answer 2

• It is a relation from a set , the domain, to a set B, the codomain (f: X -> Y) that satisfies 2 properties:
  1. each element of the domain has a mapping. (for each element of the domain x there exists a y of the codomain such that f(x) = y)
  2. each element of the domain has only one image (for each element x of the domain and y,z of the codomain, if f(x)=y and f(x)=z then y=z)

Question 3

• What is the image of x under f? How is it denoted?
• What is the range of X?
• What is the inverse image of y (pre-image of y)?

Answer 3

• it is the output of f for the input x, f(x);
• it is the set of all images of X under f;
• it the element x of the domain with y as its image, f^-1(x).

Question 4

Types of functions:
- when is a function one-to-one(injective)? (notation and definition)
- when is a function onto(surjective)? (notation and definition)
- when is a function a one-to-one correspondence (bijective)? (definition)

Answer 4

-ONE_TO_ONE: for each x1, x2 of the domain,
if f(x1) = f(x2) then x1=x2.
-> each element of the domain has a different image.

-ONTO: for each year of the codomain, there exists a x of the domain such that f(x)=y.
-> all the elements of the codomain are covered.

• if a function is both one-to-one and onto, then it is a one-to-one correspondence (bijective).

Question 5

A relation R is:
- reflexive if…
- symmetric if…
- transitive if…
- equivalent if…

Answer 5

• reflexive: if for each x of the domain, xRx.
• symmetric: for each x,y of the domain, if xRy then yRx.
• transitive: for each x,y,z of the domain, if xRy and yRz then xRz.
• equivalent if it is both reflexive, symmetric and transitive.