Functions and Relations

Question 1

What is a relation?

Answer 1

Let A and B be sets. A relation from A to B is a subset of the cartesian product A x B

i.e. A = {cities of the wold} B = {countries of the world} and R = {(a, b): a is the capital city of b} –> (Paris) R (France)

Question 2

What are Domain and Co-Domain?

Answer 2

A relation R from A to B is a subset of A x B. The set A is called domain of R and the set of B is called its co-domain

Question 3

What is a function?

Answer 3

A function from a set A to B is a relation with domain A and co-domain B that satisfies the two following properties:

1. Every element in 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 4

When are two functions equal?

Answer 4

If F:X->Y and G:X->Y are functions, then F=G if and only if F(x) = G(X) for all x in X

Question 5

What is a one to one function?(Injective)

Answer 5

A function whgere any distinct element in X is sent to a distinct element in Y (without two Xs being sent to the same element in Y)

Question 6

When is a function onto? (surjective)

Answer 6

A function is onto only if any element in Y is reached by an element x (y = F(x))

Question 7

What is a one to one correspondence (bijection)?

Answer 7

a function F:X -> Y that is both one-to-one and onto

Question 8

What is an inverse function?

Answer 8

A function that reverses the previous function

Question 9

What are the properties of a relation? (three)

Answer 9

1. Reflexive: if for all x in A, x R x
2. Symmetric: for all x and y in A, if (x, y) is part of thre relation then (y, x) has to be part of the relation as well.
3. Transitive: if for all x, y, z in A if xRy and yRz then xRz