Sets, Logic & Combinatorics

Overview

Not all AI math is continuous. Discrete mathematics — sets, logic, and counting — underlies algorithms, data structures, symbolic reasoning, and a surprising amount of practical ML plumbing. Set operations (union, intersection, difference) power deduplication, vocabulary building, and metrics like Jaccard similarity used in retrieval. Boolean logic (AND/OR/NOT, implication) is the basis of rule systems, decision-tree splits, and the conditions guardrails evaluate. Combinatorics — permutations and combinations — is how you count configurations, reason about the size of a hypothesis space, compute probabilities, and understand why brute-force search explodes (the combinatorial explosion that motivates smarter algorithms and heuristics). These tools are less glamorous than gradients but show up constantly when you build real systems around models.

Set operations power similarity and dedup

a = {"nlp", "python", "ml", "data"}
b = {"python", "ml", "cloud"}

print(a & b)                 # {'ml','python'}  intersection
print(a | b)                 # union
jaccard = len(a & b) / len(a | b)
print(round(jaccard, 3))     # 0.4  -> set-based similarity

Combinatorics: counting and the explosion

from math import comb, perm, factorial

print(comb(52, 5))       # 2,598,960 five-card poker hands (order-free)
print(perm(10, 3))       # 720 ordered arrangements of 3 from 10

# Combinatorial explosion: arrangements of n items = n!
for n in [5, 10, 15]:
    print(n, factorial(n))   # 120, 3.6M, 1.3 trillion -> why we need heuristics

Key Points to Remember

• 1Set operations (∪, ∩, −) power dedup, vocabularies, and Jaccard similarity
• 2Boolean logic underlies rule systems, decision-tree splits, and guardrail conditions
• 3Combinatorics counts arrangements (permutations) and selections (combinations)
• 4Factorial growth explains the combinatorial explosion that motivates heuristics & search