SETS AND STRUCTURES

Set

A collection of objects, called elements. Example: A={1,2,3}

Subset A⊆B

Every element of A also belongs to B. Equality is allowed

Proper subset A⊂B

Every element of A belongs to B, but B contains at least one element not in AA.

Equality of sets A=B

A=B when they contain exactly the same elements => A⊆B & B⊆A

Union of 2 sets

A ∪ B - The set containing every element of A and B

Intersection of 2 sets

A ∩ B - The set of elements belonging to both A and B

Disjoint set

2 sets with no common elements A ∩ B = ∅

Union of a family

Contains every element that belongs to at least one of the sets A

Intersection of a family

Contains every element that belongs to all the sets A

Set difference

A \ B -Elements belonging to A but not to B

Universal set U

The set of all objects being considered in the current context.

Complement sets

Ac = U \ A - All elements of the universal set that do not belong to A

De Morgan’s law: complement of an intersection

Not belonging to both means missing at least one

De Morgan’s law: complements of a union

Not belonging to either means belonging to both complements.

Closed under an operation

Applying the operation to elements of a set produces another element of the same set, whenever the operation is defined.

N - Natural numbers

All positive integers and 0

Z - integers

All integers and 0

Q - Rational numbers

All numbers expressible as p/q

R -Real numbers

All rational and irrational numbers; they form the real number line

Relationship between number systems

Density of Q in R

Between any two distinct real numbers, there is a rational number

Trichotomy in R

For any real x,y exactly one holds x<y, x=y or x>y

Total order on R

Every pair of real numbers can be compared: either X ≤ Y, X ≥ Y

Interval

A nonempty subset of R containing every number between any two of its elements

Closed interval

[ a, b ] - includes both of its endpoints

Open interval

( a, b) - Excludes both endpoints

Half-open/closed intervals

( a, b ] or [ a,b) - includes only one of its endpoints

Extended reals

R ∪ { - ∞ , ∞} often written as [ -∞ , +∞]

Upper bound of A

A real number h s.t. x ≤ h for every x ∈ A (It doesn’t need to belong to A)

Lower bound of A

A real number h s.t. x ≥ h for every x ∈ A (It doesn’t need to belong to A)

Bounded from above

A set has at least one real upper bound.

Bounded below

A set has at least one real lower bound.

Bounded set in R

A set that is both bounded above and bounded below.

Unbounded set

A set that is not bounded. It may be unbounded above, below, or both.

Maximum: maxA

An element of A greater than or equal to every element of A

Minimum: minA

An element of A less than or equal to every element of A

Bounded VS min/max

A bound may lie outside of A. A maximum or minimum must belong to A

Supremum: supA

The least upper bound of A: the smallest number that is at least as large as every element of A (It doesn’t need to belong to the set)

Infimum: infA

The greatest lower bound of A: the largest number that is at most as large as every element of A (It doesn’t need to belong to the set)

Completeness principle

Every nonempty set of R bounded above/below has a real supremum/infimum

Distance in R

d(x, y)=| x - y |

Neighborhood of x in R

Be(x) = (x - e , x + e) where e>0 Be(x) = {x: d(x,x) < e}: any open and bounded interval (a,b) with x as midpoint

Right half-neighborhood

Left half-heighborhood

Cartesian product A x B

The set of all ordered pairs ( a, b) with a ∈ A and b ∈ B

Cartesian plane: R2

All ordered pairs of real numbers. Each pair identifies a point in the plane

Cartesian space: R3

All ordered triples or real numbers

Vector

An ordered collection of components. Geometrically, it can be represented by an arrow from the origin to the corresponding point.

Parallelogram rule

When two vectors start at the same point, their sum is the diagonal of the parallelogram they form.

Scalar

A real number used to multiply a vector

Vector space

A set closed under vector addition and scalar multiplication, satisfying the addition and scalar multiplication laws.

Orthogonal vector

Vectors satisfying x*y=0: x and y form a right triangle

Weak vector inequality X ≥ Y

Xi ≥ Yi for every component

Strict vector inequality X > Y

Xi ≥ Yi for every component, with Xi > Yi for at least one component

Strong vector inequality X » Y

Xi > Yi for every component

Incomparable vectors

One vector has a larger component in one position and a smaller component in another. Neither is weakly above the other.

Convex set

For any a and b between x and y, all numbers between a and b are also an interval

Convex combination

It gives a point on the line segment joining the two vectors.

Norm

Absolute value in R (module in 2nd dimension): distance form the vector’s endpoint to the origin

Orthonormal vector

A set of orthogonal vectors each having a norm of 1

How do you make an orthogonal set orthonormal?

If every vector is nonzero, divide each vector by its own norm.

Pythagorean theorem for vectors

If x and y are orthogonal then |X + Y|2 = |x|2 + |y|2

Euclidean distance in Rn

Interior point

A point around which you can fit a sufficiently small open ball entirely inside the set and must belong to the set.

Exterior point

A point around which you can fit a sufficiently small open ball entirely outside the set and doesn’t belong to the set.

Boundary point

A point that is neither in the exterior or interior: may belong to A or Ac

Closure

The set A and all its boundary points

Isolated point

A point in the set with a neighborhood containing no other points of the set but it belongs to the set and it’s a boundary point

Limit/accumulation point

A point that has other points of A arbitrarily close to it, it doesn’t need to belong to A, all interior points are limit point

Derived set A’

The set of all limit or accumulation points of A

Boundary point outside A

Every boundary point that doesn’t belong to A is a limit point of A

Boundary points inside A

They are either isolated points or limit points.

Closure VS derived set

Closure includes all original points, including isolated points. The derived set contains only accumulation points

Open set

A set in which every point is an interior point (X,Y): if and only if A=intA, it cannot contain its boundary points

Closed set

A set containing all its boundary points and accumulation points: if and only if A=clA

Clopen set

A set that is both open and closed: the empty set, and the whole space R

Bounded set in Rn

There exists K>0 s.t. |x|<K for every x in A

Compact set

A set that is both closed and bounded