Math background

Sets

Set a collection of distinct elements:

  • \({a, b, c, d}\) - finite
  • \(\mathbb{R}\) : real numbers - infinite
  • \({x \in \mathbb{R}: cos(x) > 1/2}\) - set with restriction

Notations:

  • \(\Omega\) - universal set(set that contain all objects)
  • \(\in\) - is member of
    • \(2 \in \{1, 2, 3, 4\}\)
  • \(\notin\) - is not member of
    • \(2 \notin \{1, 3, 5\}\)
  • \(A’\), \(A^c\), \(\overline{\rm A}\) - \(A\) compliment, everything not in \(A\)
    • \(x \in S^c if x \in \Omega, x \notin S\)
  • \(\emptyset\) and \(\varnothing\) or \(\{ \}\) - empty set
    • \(\Omega^c = \emptyset\)
  • \(\cup\) - union, everything in both sets
    • \(\{1, 2, 3\} \cup \{3, 4, 5\} = \{1, 2, 3, 4, 5\}\)
    • \(x \in S \cup T \leftrightarrow x \in S \, \textsf{or} \> x \in T\)
    • \(\cup_{n} S_{n}\) - union of all \(n\) sets
  • \(\cap\) - intersection, only what is in common in both sets
    • \(\{1, 2, 3\} \cap \{2, 3, 4\} = \{2, 3\}\)
    • \(x \in S \cap T \leftrightarrow x \in S \, \textsf{and} \, x \in T\)
    • \(\cap_{n} S_{n}\) - intersection of all \(n\) sets
  • \(\subset\) - subset of set
    • \(\{1, 2, 3\} \subset \{1, 2, 3, 4, 8\}\)
  • \(\supset\) - superset of set
    • \(\{1, 2, 3, 4, 8\} \supset \{1, 2, 3\}\)
/images/math/math-notations/set_notations.jpg

Sets properties

  • \(S \cup T = T \cup S\)
  • \(S \cap (T \cup U) = (S \cap T) \cup (S \cap U)\)
  • \((S^c)^c = S\)
  • \(S \cup \Omega = \Omega\)
  • \(S \cap S^c = \emptyset\)
  • \(S \cap \Omega = S\)
  • \(S \cup (T \cup U) = (S \cup T) \cup U\)
  • \(S \cup (T \cap U) = (S \cup T) \cap (S \cup U)\)

De Morgan’s laws

  • \((S \cap T)^c = S^c \cup T^c\)
  • \((\cap_{n} S_{n}^c) = \cup_{n} S_{n}^c\)
  • \((S \cup T)^c = S^c \cap T^c\)
  • \((\cup_n S_{n}^c) = \cap_n S_{n}^c\)

Countable versus uncountable infinite sets

Countable: can be put in 1-1 correspondence with positive integers:

  • Positive integers
  • Integers: \(0, 1, -1, 2, -2, …\)
  • Pairs of positive integers
  • Rational numbers \(q\), with \(0 < q < 1\), \(1/2, 1/3, 2/3, 1/4, 2/4, …\)

Uncountable: not countable:

  • The interval [0, 1]
  • The reals
  • The plane

Sequences ant their limits

Sequence is some collection of an elements:

  • \(a_1, a_2, a_3, …\)
  • sequence \(a_1\)
  • \(\{a_i\}\)
  • formal - function \(f: \mathbb{N} -> S, f(i) = a_i\)

When does a sequence converge:

  • If \(a_i \leq a_{i + 1}\) for all \(i\), then either:
    • the sequence “converges to \(\infty\)
    • the sequence converges to some real number \(a\)
  • if \(|a_i - a| \leq b_i\) for all \(i\), and \(b_i -> 0\), then \(a_i -> a\)

Series

Infinite series

\(\sum_{i=1}^{\infty} a_i = \lim_{n -> \infty} \sum_{i=1}^n a_i\) (provided limit exists)

  • if \(a_i \geq 0\): limit exists
  • if terms \(a_i\) do not all have the same sign:
    • limit need not exist
    • limit may exist but be different if we sum in different order
    • Fact: limit exists and independent of order of summation if \(\sum_{i=1}^{\infty} |a_i| < \infty\)

Order of summation in series

only if \(\sum |a_{ij}| < \infty\), then:

  • \(\sum_{i \geq 1, j \geq 1} a_{ij} = \sum_{i=1}^{\infty}(\sum_{j = 1}^{\infty} a_{ij}) = \sum_{j=1}^{\infty}(\sum_{i = 1}^{\infty} a_{ij})\)
  • \(\sum_{(i, j) : j \leq i} a_{ij} = \sum_{i=1}^{\infty} \sum_{j=1}^{i} a_{ij} = \sum_{j=1}^{\infty} \sum_{i=j}^{\infty} a_{ij}\)

Arithmetic progression

\begin{equation*} a_{n}=a_{1}+(n-1)d \end{equation*}
\begin{equation*} a_{n}=a_{m}+(n-m)d \end{equation*}
\begin{equation*} S_n = \frac{n}{2}[2a_1 + (n - 1)d] \end{equation*}
\begin{equation*} S_n = \frac{n}{2}[a_1 + a_n] \end{equation*}
\begin{equation*} \overline{\rm n} = S_n / n \end{equation*}
\begin{equation*} \overline{\rm n} = \frac{a_1 + a_n}{2} \end{equation*}

where:

  • \(a_1\) is the first term of an arithmetic progression.
  • \(a_n\) is the nth term of an arithmetic progression.
  • \(d\) is the difference between terms of the arithmetic progression.
  • \(n\) is the number of terms in the arithmetic progression.
  • \(S_n\) is the sum of n terms in the arithmetic progression.
  • \(\overline{\rm n}\) is the mean value of arithmetic series.

Geometric progression

\begin{equation*} a_n = ar^{n - 1} \end{equation*}
\begin{equation*} a_n = ra_{n - 1} \end{equation*}
\begin{equation*} S_n = \frac{a_1 (1 - r^n)}{1 - r} \end{equation*}
\begin{equation*} S_n = \frac{a_1 - a_n * r}{1 - r} \end{equation*}
\begin{equation*} where r \ne 1 \end{equation*}

where:

  • \(a\) - initial value
  • \(r\) - common ratio

Simplified: \(S = \sum_{i=0}^{\inf} \alpha^i = 1 + \alpha + \alpha^2 + … = \frac{1}{1 - \alpha}\), if \(|\alpha| < 1\)

pass

Other

If \(X\) is a vector, and \(X \in \{-1, 1\}^{n}\), then \(X^{T}X == n\), where \(n\) is a constant, and \(n == len(X)\).