Miscellaneous

Summation notation

Rules of sums & Newton's binomial formula

Useful formulas

  1. \(x=1+2+...+(n-1)+n\)
  2. We can rewrite this as \(x=n+(n-1)+...+2+1\)
  3. From these we can write \(2x=[n+(n-1)+...+2+1]+[1+2+...+(n-1)+n]\)
  4. Rearranging we get \(2x=(n+1)+(n-1+2)+...+(2+n-1)+(1+n)=(n+1)+(n+1)+...+(n+1)+(n+1)\)
  5. Finally, we get \(2x=n\left(n+1\right)\). Solving for \(x\)\(x=\frac{1}{2}n(n+1)\). QED
  1. \((a+b)^3=a^3+\left( \begin{array}{c} 3 \\ 1 \end{array} \right)a^{2}b+\left( \begin{array}{c} 3 \\ 2 \end{array} \right)ab^{2}+b^3\). We can use the formula from above to write \((a+b)^3=a^3+\frac{3}{1}a^{2}b+ \frac{3\cdot 2}{1\cdot 2}ab^{2}+b^3=a^3+3a^2b+3ab^2+b^3\)

Double sums

A few aspects of logic

  1. If \(P \Longrightarrow Q\) we say that p is sufficient condition for q
  2. If \(P\Longrightarrow Q\) we say that q is necessary condition for p

Solving equations

Mathematical proofs

  1. Direct proof: We start from the premises and we keep deriving their implications until we get to the conclusions.
  2. Indirect proof: We deny the conclusions and show that the premises must also be false. The indirect proof relies on the fact that \(P\Longrightarrow Q\) is equivalent to \(\backsim Q \Longrightarrow \backsim P\) (non-q implies non-p)

Essentials of set theory

Property of a set

Set membership

  1. \(\subset\)
  2. \(\subseteq\)
  3. \(\in\)
  4. \(\notin\)

Set operations

  1. Union\(\cup \). All the elements that belong to at least one of the sets.
  2. Intersection\(\cap\). All the elements that belong to both sets.
  3. Minus\(\setminus\) .All the elements that belong to one set but not the other.
  1. Disjoint set: the empty set. For example, the intersection between two sets that do not share any members
  2. The universal set\(\Omega\): The set containing all potential subsets of a family of sets.
  3. Complement set\(A^c=\Omega\setminus A\). It contains all elements of the universal set not contained in A.