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Sets and Functions

Some important symbols

Symbol Used for
\(\implies\) Implies
\(\forall\) For All
\(\exists\) There exists
\(\Leftrightarrow\) Equivalence

Sets

Some basic sets to remember

  • $\R \to $ Set of Real Numbers
  • $\R_+ \to $ Set of Positive Real numbers (including \(0\))
  • \(\Z \to\) Set of all integers
  • \(\Z_+ \to\) Set of positive integers including \(0\)
  • \([a,b] \to \{x\in\R\ |\ a\leq x\leq b\}\)
  • \((a,b) \to \{x\in\R\ |\ a< x< b\}\)
  • $\R^d \to $ Set of \(d\)-dimensional vectors
  • \([a,b]^d \to \{x\in\R^d:x_i\in[a,b], i\in\{1,2,3,\dots d\}\}\)

Metric Spaces

Metric spaces are basically vectors with some Distance function \(D\) associated with them.

For most of question throughout the course the default metric space used is \(\R^d\) where \(D=\sqrt{(x_1-y_1)^2+(x_2-y_2)^2+\dots(x_d-y_d)^2}\),

Open ball in Metric Space

Let \(x\in\R^d\). Then, the open ball is given by \(B(x,\epsilon)=\{y\in\R^d: D(x,y)<\epsilon\}\)

Closed ball in Metric Space

Let \(x\in\R^d\). Then, the closed ball is given by \(B(x,\epsilon)=\{y\in\R^d: D(x,y)\leq\epsilon\}\)

Tip

If you include the boundary, it is a closed ball, else an open ball.

Some basic operations defined on sets are

  • Union (\(\cup\))
  • Intersection (\(\cap\))
  • Complement (\(S^{c}\))
  • \(A-B\)

Union (\(\cup\))

The union of two sets \(A\) and \(B\) is the set of elements which are in \(A\), as well as in \(B\) or in both.

Intersection (\(\cap\))

The intersection of two sets \(A\) and \(B\) is the set of elements which are in both \(A\) and \(B\).

Complement (\(S^{c}\))

The complement of a set \(S\) is the set of all elements which are in the universal set but not in \(S\).

\(A-B\)

This contains all the elements of \(A\) that are not in \(B\).

De Morgan's Law

Let there be two sets \(A\) and \(B\). Then, the De-Morgan's Law states that,

  • \((A\cup B)^{c}=A^c\cap B^c\)
  • \((A\cap B)^{c}=A^c\cup B^c\)

Sequences

An ordered collection of elements is known as a sequence.

Let there be a sequence where the elements are \(x_1,x_2,...\), where \(x_i\in\R^d\). And, if \(lim_{i\to\infin} x^i = x^*\) then, \(\forall\text{ } \epsilon>0, \exists \text{ a }N, \text{ such that } x_n\in B(x^*, \epsilon)\ \forall\ n\geq N\)

Vector Spaces

A vector space is a set of elements called vectors, having operations of addition and scalar multiplication defined on it.

A vector space must satisfy the following properties:

  • Additive Closure: If the vector space is \(V\) and \(v_1,v_2\in V\), then \(c_1v_1 + c_2v_2\in V\), where \(c_1,c_2\in \R\)
  • Multiplative Closure

Dot Product of two vectors

Let \(x\) and \(y\) be two vectors such that \(x,y\in\R^d\). Then, \(x.y=x^Ty=\sum_{i=1}^dx_i.y_i\)

Norm of a vector

Let \(x\) be a vector such that \(x\in\R^d\). Then, the norm of the vector is given by \(||x||=\sqrt{x^Tx}=\sqrt{\sum_{i=1}^dx_i^2}\)

Orthognal Vectors

Let \(x\) and \(y\) be two vectors such that \(x,y\in\R^d\). Then, \(x\) and \(y\) are orthogonal if \(x.y=0\), i.e., \(x^Ty=0\)

Functions And Graphs

Let \(f\) be a function with domain \(A\) and co-domain \(B\). Then, the function is denoted as \(f:A\to B\).

\(d\)-dimensional functions

A function with domain \(\R^d\) to \(\R\) is known as a \(d\)-dimensional function or multivariate function.

It is denoted as \(f:\R^d\to\R\)

Graph of a function \(f\)

The graph of a \(d\)-dimensional function is defined as \(G_f \subseteq \R^{d+1}\).

\[ G_f=\{(x, f(x)): x\in\R^d\} \]