Lesson 1

We use the following standard notation and facts.

• Notation 
  • \([P]\) is the Iverson bracket: \([P]=1\) if \(P\) is true, and \([P]=0\) otherwise (\([2<3]=1\), \([2 \le 3]=1\), \([2>3]=0\))
  
  
  • \(\lfloor x \rfloor\) is the floor of \(x\), that is, rounding \(x\) down to the nearest integer (\(\lfloor 3 \rfloor=3\), \(\lfloor 3.01 \rfloor=3\), \(\lfloor 3.873 \rfloor=3\))
  
  
  • \(\sum_{i \in [n]}f(i)=f(1)+f(2)+\dotsb+f(n)\) is the sum of \(f(i)\) over all \(i\) from \(1\) to \(n\)
  
  
  • \(a \equiv_{m} b\) states that the integers \(a\) and \(b\) are congruent modulo \(m\) (i.e., they give the same remainder when divided by \(m\); equivalently, their difference is divisible by \(m\))
  
  

• Sets 
  • \([n]=\{1,2,\dotsc,n\}\) is the set of positive integers from \(1\) to \(n\)
  
  
  • \(\mathbb{Z}=\{\dotsc, -2, -1, 0, 1, 2, \dotsc\}\) is the set of all integers
  
  
  • \(\mathbb{Z}_{\ge n}=\{n, n+1, n+2, \dotsc\}\) is the set of integers not less than \(n\)
  
  
  • \(\mathbb{P}=\{2,3,5,7,11,13,17,19,23,\dotsc\}\) is the set of prime numbers
  
  
  • \(\mathbb{R}\) is the set of real numbers
  
  

• Inequalities 
  • \(1+x \le e^{x}\) for all \(x \in \mathbb{R}\)
  
  

• Set theory 
  • \(|A|\) is the size (or cardinality) of the set \(A\)
  
  
  • \(x \in A\) states that the element \(x\) belongs to the set \(A\) (or the set \(A\) contains the element \(x\))
  
  
  • \(A \cup B\) is the union of the sets \(A\) and \(B\), i.e., all elements that lie in at least one of the sets \(A\) and \(B\)
  
  
  • \(A \cap B\) is the intersection of the sets \(A\) and \(B\), i.e., all elements that lie in both sets \(A\) and \(B\)