Discrete Math Flashcards
Free discrete math flashcards on logic, sets, functions, counting, recurrences, graph theory and proof techniques. Flip each card, mark what you know and drill what you miss for your intro discrete math course.
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All 45 cards at a glance
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Logic
| Question | Answer |
|---|---|
| What is a proposition? | A statement that is either true or false. |
| Conjunction p ∧ q is true when? | Both p and q are true. |
| Disjunction p ∨ q is true when? | At least one of p and q is true. |
| Negation ¬p? | True exactly when p is false. |
| Implication p → q is false when? | p is true and q is false. It is true in all other cases. |
| Contrapositive of p → q? | ¬q → ¬p, which is logically equivalent to p → q. |
| Converse of p → q? | q → p, which is not equivalent to p → q. |
| De Morgan’s laws? | ¬(p ∧ q) ≡ ¬p ∨ ¬q and ¬(p ∨ q) ≡ ¬p ∧ ¬q |
| What does ∀ mean? | For all (universal quantifier). |
| What does ∃ mean? | There exists (existential quantifier). |
| Negation of ∀x P(x)? | ∃x ¬P(x) |
Sets and functions
| Question | Answer |
|---|---|
| Union A ∪ B? | Elements in A or B (or both). |
| Intersection A ∩ B? | Elements in both A and B. |
| Complement of A? | Elements of the universal set not in A. |
| Power set of a set with n elements has how many elements? | 2ⁿ |
| Inclusion-exclusion for two sets? | |A ∪ B| = |A| + |B| − |A ∩ B| |
| Injective (one-to-one) function? | Different inputs give different outputs. |
| Surjective (onto) function? | Every element of the codomain is an output. |
| Bijective function? | Both injective and surjective, so it has an inverse. |
| Cartesian product A × B? | The set of all ordered pairs (a, b) with a in A and b in B. |
Counting
| Question | Answer |
|---|---|
| Product rule? | If tasks are done in sequence with m and n choices, there are m × n ways. |
| Sum rule? | If two tasks cannot both be done, with m and n ways, there are m + n ways in total. |
| Permutations of n distinct objects? | n! |
| r-permutations of n objects? | n! / (n − r)! |
| r-combinations of n objects? | n! / [r!(n − r)!] |
| Pigeonhole principle? | If more than k items are placed in k boxes, some box has at least two items. |
| Number of subsets of an n-element set? | 2ⁿ |
| Binomial coefficient identity (Pascal)? | C(n, k) = C(n−1, k−1) + C(n−1, k) |
Graph theory
| Question | Answer |
|---|---|
| Graph? | A set of vertices and a set of edges connecting pairs of vertices. |
| Degree of a vertex? | The number of edges attached to it. |
| Handshake theorem? | The sum of all vertex degrees equals twice the number of edges. |
| Tree? | A connected graph with no cycles. |
| Number of edges in a tree with n vertices? | n − 1 |
| Euler circuit exists when? | The graph is connected and every vertex has even degree. |
| Bipartite graph? | A graph whose vertices split into two sets with edges only between the sets. |
| Planar graph? | A graph that can be drawn in the plane with no crossing edges. |
| Euler’s formula for connected planar graphs? | V − E + F = 2 |
Proofs and recurrences
| Question | Answer |
|---|---|
| What is proof by contradiction? | Assume the statement is false and derive a contradiction. |
| What is proof by induction? | Prove a base case, then prove that if it holds for k it holds for k + 1. |
| What is a direct proof? | Assume the hypothesis and reason step by step to the conclusion. |
| What is a counterexample? | One example that shows a universal statement is false. |
| Fibonacci recurrence? | Fₙ = Fₙ₋₁ + Fₙ₋₂, with F₁ = F₂ = 1. |
| What is a recurrence relation? | A rule that defines each term of a sequence using earlier terms. |
| What does a mod b mean? | The remainder when a is divided by b. |
| GCD by the Euclidean algorithm? | Repeatedly replace (a, b) with (b, a mod b) until the remainder is 0. |
Frequently asked questions
What is discrete math used for?
It underlies computer science: algorithms, data structures, cryptography, databases and logic all use discrete math.
Is discrete math hard?
It is more about reasoning and proofs than computation, so many students find the proof techniques the hardest part.
Sources
More free math flashcards
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