Discrete Math for Data Science

DSCI 220, 2025 W1

October 6, 2025

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Modular Arithmetic

Modular Arithmetic

Definition: \(a \equiv b \pmod m \iff m \mid (a-b)\)

If \(a \equiv b \pmod m\) and \(c \equiv d \pmod m\), then:

• Addition: \(a + c \equiv b + d \pmod m\)
• Subtraction: \(a - c \equiv b - d \pmod m\)
• Multiplication: \(ac \equiv bd \pmod m\)
• Exponentiation: for any integer \(n \ge 0\), \(a^{n} \equiv b^{n} \pmod m\)

Reduce anytime: reduce intermediate results mod \(m\) at any step.

Caution (division): “Division” is only valid when a modular inverse exists.
\(\gcd(a,m)=1\Rightarrow a^{-1}\) exists, and \(ax \equiv b \pmod m \Rightarrow x \equiv a^{-1}b \pmod m\).

Caution (exponents): From \(a\equiv b \pmod m\) you may raise the bases: \(a^n\equiv b^n\), but in general \(c^{a}\not\equiv c^{b}\pmod m\).

Inverse \(\bmod m\)

When does a number \(a\) not have an inverse \(\bmod m\)?

Consider \(5x \bmod 12\):

Consider \(9x \bmod 12\):

 

A number \(a\) has an inverse \(\bmod m\) iff \(\gcd(a,m)= 1\).

Linear Congruences

 

Solve:

• \(9x\equiv 1\pmod{26}\)

 

 

• \(9x\equiv 5\pmod{26}\)

Big Powers Fast

 

Pick one:

• \(2^{1000}\pmod 7\)
  
• \(3^{12345}\pmod{10}\)
  
• \(5^{2024}\pmod{13}\)

Timing: ~6–8 min. 1) \(2^{3}\equiv1\Rightarrow 2^{1000}\equiv 2\). 2) Period \(4\) mod \(10\); \(12345\equiv1\Rightarrow 3\). 3) \(5^2\equiv-1\Rightarrow 5^4\equiv1\); \(2024\equiv0\pmod4\Rightarrow 1\).

Data-science tie: “Cycle detection helps compute modular hashes fast in streaming.”

Two Dials

Find \(t\pmod {15}\) such that
\(t\equiv 2\pmod 3\) and \(t\equiv 1\pmod 5\).

05:00

Timing: ~6–8 min. Numbers \(\equiv1\pmod5\): \(1,6,11,\dots\); mod \(3\) these are \(2, 5, 8, 11\dots\) → pick \(11\). Unique mod \(15\).

Asymptotics

Functions

Math: a function is a mechanism for translating items from one collection to another.

 

\(f: A\to B\)

Our current interest is quantifying the way that functions grow.

there are many different ways we can talk about functions… may do so later. In the course of this discussion I’ll mention some other relevant features, but they will not be our focus.

Things I WILL mention: domain, co-domain (range), every element from domain is mapped exactly once to some element from co-domain.

Functions for Data Science

 

Functions as costs:

1. generally non-negative.

2. generally non-decreasing with size of input.

Sketch:

Asymptotic Upper Bounds

A function \(f(n) = O(g(n))\) iff there are positive constants \(c\) and \(n_0\) such that \(f(n)\leq c\cdot g(n)\) for any \(n\geq n_0\).

 

Big O Proof

Prove that \(3n^2 + 7n + 2 = O(n^2)\).

Defn: … there are positive constants \(c\) and \(n_0\) such that \(f(n)\leq c\cdot g(n)\) for any \(n\geq n_0\).

 

 

 

 

Therefore, \(c=\)____ and \(n_0=\)____ witness the claim.

Big O Strategy

Goal. To prove \(f(n)=O(g(n))\), show there exist constants \(c>0\) and \(n_0\) such that
for all \(n\ge n_0\), \(f(n)\le c\,g(n)\).

1. Pick your region.
   Choose \(n_0\) where simple comparisons are true (e.g., for \(n\ge 1\), \(n^2\ge n\); for \(n\ge e\), \(\log n\ge 1\)).

2. Dominate each term by \(g(n)\).
   On \(n\ge n_0\), bound every term of \(f\) by a constant multiple of \(g\):
   \(f(n)=\text{(term}_1)+\dots+\text{(term}_k)\ \le\ a_1 g(n)+\dots+a_k g(n)\).

3. Add and name the constant.
   Sum the multipliers: \(f(n)\le (a_1+\cdots+a_k)\,g(n)\).
   Set \(c=a_1+\cdots+a_k\).

4. State the line that matters.
   “For all \(n\ge n_0\), \(f(n)\le c\,g(n)\). Therefore \(f(n)=O(g(n))\).”