Bocconi Mathematics: Theorems & definitions for the 1st math partial (Unit 1 Structures): -

Theorem on Injectivity and Invertibility:

Given f: A c_ R, function f is invertible on A if and only if f is injective

Definition of Subsets:

Given two sets A and B, we say that A is a subset of B, in symbol A c_ B, if all the elements of A are also elements of B, that is, if X 'belongs to' A implies X 'belongs to' B

Definition of the Empty Set:

The empty set denoted by O/ is the set w/o elements

Definition of Intersection:

Given two sets A and B, their intersection A n B is the set of all the elements that belong to A and B, that is X 'belongs to' A n B if X 'belongs to' A and X 'belongs to' B

Definition of Union:

Given two sets A and B, their union A u B is the set of all the elements that belong to A or to B, that is, X 'belongs to' A u B, if X 'belongs to' A or X 'belongs to' B

Definition of Difference:

Given two sets A and B, their difference A\B is the set of all the elements that belong to A, but not to B, that is X 'belongs to' A\B if both X 'belongs to' A and X 'doesn't belong to' B.

Proof of first De Morgan's Law Method 1:

Law: (A u B)c C_ Ac n Bc

Proof:

(A u B)c C_ Ac n Bc;

if X 'belongs to' (A u B)c => X 'doesn't belong to' A u B

=> X 'doesn't belong to' A OR X 'doesn't belong to' B

=> X 'belongs to' Ac AND X 'belongs to' Bc

=> X 'belongs to' Ac n Bc

Proof of first De Morgan's Law Method 2 (other way around):

Law: (A u B)c C_ Ac n Bc

Proof:

Ac n Bc C_ (A u B)c;

if X 'belongs to' Ac n Bc => X 'belongs to' Ac AND X 'belongs to' Bc

=> X 'doesn't belong to' A OR X 'doesn't belong to' B

=> X 'doesn't belong to' A u B

=> X 'belongs to' (A u B)c

Proof of second De Morgan's Law Method 1:

Law: (A n B)c C_ Ac u Bc;

Proof:

if X 'belongs to' (A n B)c => X 'doesn't belong to' A n B;

=> X 'doesn't belong to' A AND X 'doesn't belong to' B

=> X 'belongs to' Ac OR X 'belongs to' Bc

=> X 'belongs to' Ac u Bc

Proof of second De Morgan's Law Method 2 (other way around):

Law: (A n B)c C_ Ac u Bc;

Proof:

Ac u Bc C_ (A n B)c

if X 'belongs to' Ac u Bc => X 'belongs to' Ac OR X 'belongs to' Bc

=> X 'doesn't belong to' A AND X 'doesn't belong to' B

=> X 'doesn't belong to' A n B

=> X 'belongs to' (A n B)c

Definition of Upper Bound:

Let A c_ R be a non-empty set. A number h 'belonging to' R is called 'upper bound' of A, if it is greater than or equal to each element of A, that is, if h _> X, 'for all' X 'belonging to' A

Definition of Lower Bound:

Let A c_ R be a non-empty set. A number h 'belonging to' R is called 'lower bound' of A if it is smaller than or equal to each element of A, that is, if h <_ X, 'for all' X 'belonging to' A

Definition of the boundary conditions of a non-empty set A c_ R:

A non-empty set A c_ R is said to be:

i) bounded (from) above if it has an upper bound that is A* =/ O/

ii) bounded (from) below if it has a lower bound, that is A*(star at bottom) =/ O/

iii) bounded if its bounded (from) both above and below

Definition of Maximum:

Given a non-empty set A c_ R, an element X^ of A is called "maximum" of A if it is the greatest element of A, that is, if X^ _> X, 'for all' X 'belonging to' A.

Definition of Minimum:

Given a non-empty set A c_ R, an element X^ of A is called "minimum" of A if it is the smallest element of A, that is, if X^ <_ X, 'for all' X 'belonging to' A

Definition of Supremum and Infinum:

Given a non-empty set A c_ R, the supremum of A is the least upper bound of A, that is, min. A (star at top), while the infinum is the greatest lower bound of A, that is max. A (star at bottom)

(sup A = min. A; inf A = max. A (star at bottom)

Theorem of the Least Upper Bound Principle:

Each non-empty set A c_ R has a supremum if it is bounded above and it has an infinum if it is bounded below.

Definition of Absolute Value:

'for all' X 'belonging to' R, geometrically it represents the distance of X from the origin, that is 'for all' X 'belonging to R' |X| = dis (X, 0)

Definition of Distance:

Let X, Y 'belong to' R, w/ X < Y => distance of y from x is defined by the difference Y - X => dis (X,Y) = Y - X

(If we don't know the order of X and Y we write: dis (X,Y) = |X - Y| where |X-Y| -> X - Y if X _> Y; Y - X if X < Y)

Definition of Neighborhoods:

Given 𝜺 > 0, the interval [X0, X0 + 𝜺] is called the right neighborhood of X0 'belonging to' R of radius 𝜺 > 0. The interval [X0 - 𝜺, X0] is called the left neighborhood of X0 of radius 𝜺 > 0.

Definition on Neighborhoods & Half-lines:

A neighborhood of +infinity is a half-line [k, +infinity), with k 'belonging to' R.

A neighborhood of -infinity is a half-line (-infinity, k] with k 'belonging to' R

Definition of the Cartesian Product:

Given two sets A1 and A2, the cartesian product A1 x A2 is the set of all order pairs (a1, a2) w/ a1 'belonging to' A1, a2 'belonging to' A2

Definition of the Cartesian Product of n Sets:

Given n sets, A1, A2, ... An their cartesian product A1 x A2 x ... x An, denoted by n π i=1 Ai, is the set of all the ordered n-tuples (a1, a2, ..., an) with a1 'belonging to' A1, a2 'belonging to' A2, ... , an 'belonging to' An

Definition of Linear Combination:

A vector X_ 'belonging to' R^n is said to be a linear combination of the vectors {X_1, X_2, ..., X_n} of R^n if there exist n scalars {𝛼1, 𝛼2, ..., 𝛼n] such that, X_ = 𝛼1 X_1 + 𝛼2 X_2 + ... + 𝛼n X_n ('belonging to' R^n)

Definition of a Convex Set:

A set C of R^n is said to be convex if, for every pair of points X_, Y 'belonging to' C, 𝛼X + (1 - 𝛼)Y_ 'belongs to' C, 'for all' 𝛼 'belonging to' [0,1]

Definition on Euclidean Distance:

The euclidean distance d(X, Y) between two vectors X_ and Y_ in R^n is the norm of their difference; d(X_ , Y_) = ||X_ - Y_||

Definition on Neighborhoods of a Center w/ Radius:

A neighborhood of center X_0 'belonging to' R^n and w/ radius 𝜺 > 0, denoted by B𝜺 (X_0), is the set

B𝜺 (X_0) = {X_ 'belongs to' R 'such that' d(X_ , X_0) < 𝜺}

Definition of Taxonomy of the Points of R^n:

Let A be a set in R^n. A point X_0 'belonging to' R^n is an interval point of A if there exists 𝜺 > 0 such that B𝜺 (X_0) c_ A.

Definition of boundary points in R^n:

Let A be a set in R^n. A point X_0 'belonging to' R is a boundary point of A if it is neither interior nor exterior, i.e., if for every 𝜺>0 both B𝜺 (X_0) n A =/ O/ and

B𝜺 (X_0) n Ac =/ O/

(Boundary points belong to both A & Ac)

Definition of isolated points in R^n:

Let A be a set in R^n. A point X_0 'belonging to' A is isolated if there exists a neighborhood B𝜺 (X_0) of X_0 that does not contain other points of A except for X_0 itself, i.e. A n B𝜺 (X_0) = {X_0}

Definition of limit points:

Let A be a set in R^n. A point X_0 'belonging to' R^n is called a limit (or accumulation) point of A if each neighborhood B𝜺(X_0) of X_0 contains at least one point of A distinct from X_0.

So, X_0 is a limit point of A if 'for all' neighborhoods B𝜺 (X_0), 'there exist at least one' X_1 = X_0

Definition of an Open Set:

A set A in R^n is called "open" if all its points are interior, that is, if int. A = A. (A set is open if it does not contain its borders (it is skinless))

Lemma on neighborhoods and open sets & PROOF:

Neighborhoods are open sets.

PROOF: Let B𝜺 (X_0) be a neighborhood of X_0 'belonging to' R. We have to show that all its points are interior.

Let X_ 'belong to' B𝜺 (X_0). This means d (X_ , X_0) < 𝜺', so we can write 'there exists at least one' 𝜺' _> 0:

0 < 𝜺' < 𝜺 - d(X_ , X_0)

Now, let y 'belong to' B𝜺' (X_), then

d (Y_ , X_0) <_ d (Y_ , X_) + d (X_, X_0) < 𝜺' + d (X_ , X_0) < 𝜺

=> Y_ 'belongs to' B𝜺 (X_0),

We can conclude: B𝜺' (X_) c_ B𝜺 (X_0)

Definition of a Closed Set:

A set A in R^n is called "closed" if it contains all its boundary points, that is A = A u 'frontier' A. So, A is closed when it incl. its borders.

Theorem on open and closed sets (and the subsequent state of their complements):

A set A in R^n is open if and only if (<=>) its complement is closed.

A open <=> Ac is closed

Corollary on closure of sets depending on limit points and proof:

A set in R^n is closed if and only if (<=>) it contains all its limit points.

Proof:

A being closed means A- (line above) = A, so by the preceding result A - (line above) = A u A', then we have that: A u A' = A- (line above) = A, so A' c_ A.

If A' c_ A => A u A' = A and again, A = A u A' = A^- that is, A is closed

Definition of a Bounded Set:

A set A in R^n is bounded if there exists k > 0:

||X_|| < k, 'for all' X_ 'belonging to' A

(So, A is bounded if: d(X_ , 0_) < k, 'for all' X_ 'belonging to' A)

Definition of a Compact Set:

A set A in R^n is called "compact" if it is both closed and bounded. (This definition of compactness holds only in R^n).