De Morgan’s laws
$$\neg(P\land Q)\equiv\neg P\lor\neg Q,\qquad\neg(P\lor Q)\equiv\neg P\land\neg Q$$
Move a negation through AND or OR while switching the connective.
UniversityDiscrete math formulas, grouped by topic.
11 discrete math formulas across 6 topics, free and without an account. Discrete-mathematics formulas count and characterize finite structures. Logic and set identities rewrite statements, summation formulas close off repeated addition, and graph relations tie vertices, edges, and degrees together. Because the objects are finite, most results can be checked directly on a small case — which is exactly how to confirm you applied the formula correctly.
Every discrete math formula on one page
Each entry links to its own page, where the symbols are defined, the conditions are stated, and the formula is substituted through a worked example. Levels covered here: High school, University.
Logic
1 formula in logic.
Sets
1 formula in sets.
Two-set inclusion–exclusion
$$|A\cup B|=|A|+|B|-|A\cap B|$$
Count a union without double-counting the overlap.
High school · UniversitySums
2 formulas in sums.
Sum of first integers
$$\sum_{k=1}^{n}k=\frac{n(n+1)}2$$
Add consecutive positive integers efficiently.
High school · UniversitySum of squares
$$\sum_{k=1}^{n}k^2=\frac{n(n+1)(2n+1)}6$$
Add squares of the first $n$ positive integers.
High school · UniversityGraphs
4 formulas in graphs.
Handshaking lemma
$$\sum_{v\in V}\deg(v)=2|E|$$
Relate all vertex degrees to the number of edges in an undirected graph.
UniversityEuler formula for planar graphs
$$|V|-|E|+|F|=2$$
Relate vertices, edges, and faces in a connected planar embedding.
UniversityEdges in a complete graph
$$|E(K_n)|=\binom n2=\frac{n(n-1)}2$$
Count edges joining every pair of distinct vertices.
High school · UniversityEdges in a tree
$$|E|=|V|-1$$
Relate edge and vertex counts in a finite tree.
UniversityNumber theory
2 formulas in number theory.
Euclidean algorithm
$$\gcd(a,b)=\gcd(b,a\bmod b)$$
Reduce a greatest-common-divisor problem to smaller remainders.
High school · UniversityModular congruence
$$a\equiv b\pmod n\iff n\mid(a-b)$$
State that two integers have the same remainder modulo $n$.
UniversityCounting
1 formula in counting.
Pigeonhole principle
$$N\text{ objects in }k\text{ boxes}\implies\text{some box has at least }\left\lceil\frac Nk\right\rceil\text{ objects}$$
Guarantee a repeated placement without identifying which box receives it.
High school · UniversityKeep going with discrete math
A formula sheet is a reference, not a method. Use the subject guide to see where these relationships come from, a calculator to check a substitution you are unsure about, and practice questions to find out whether you can actually retrieve them.
- Study discrete math — the subject hub with lessons, worked methods, and common mistakes.
- Discrete math calculator — enter your own values and compare the result with your working.
- Discrete Math Practice — retrieval practice across every topic listed above.
- Math formula library — search all 215 formulas by name, symbol, or condition.
Formulas in the other subjects
Mathematics does not stop at a subject boundary, and neither do its formulas. Each subject page below lists its own relationships, conditions, and worked examples.
Arithmetic
Review the core arithmetic relationships, their restrictions, and worked substitutions.
Algebra
Review the core algebra relationships, their restrictions, and worked substitutions.
Geometry
Review the core geometry relationships, their restrictions, and worked substitutions.
Trigonometry
Review the core trigonometry relationships, their restrictions, and worked substitutions.
Precalculus
Review the core precalculus relationships, their restrictions, and worked substitutions.
Calculus
Review the core calculus relationships, their restrictions, and worked substitutions.
Statistics
Review the core statistics relationships, their restrictions, and worked substitutions.
Probability
Review the core probability relationships, their restrictions, and worked substitutions.
Linear Algebra
Review the core linear algebra relationships, their restrictions, and worked substitutions.
Differential Equations
Review the core differential equations relationships, their restrictions, and worked substitutions.