Discrete mathematics assignment help works best when it teaches you to write proofs, not just reach answers. Most discrete maths marks are for a complete, logical argument.
This guide covers logic, sets and functions, proof techniques, counting, recurrences and graph theory. Each section shows what markers expect, with worked examples, and the guide ends with how STEM Donkey prepares custom solutions for study and reference.
What Discrete Maths Markers Are Checking
Discrete maths is marked on rigor. Markers check that you state what you are proving, choose a suitable proof method, justify every step and reach the claim exactly, with no gaps or circular reasoning.
A correct number for a counting problem earns little without an explanation of why the counting method fits.
Topics Discrete Maths Assignments Cover
| Topic | Typical questions | Key skill |
|---|---|---|
| Propositional logic | Truth tables, equivalences, normal forms | Applying laws such as De Morgan's |
| Predicate logic | Quantifiers, negating statements, translating English | Handling "for all" and "there exists" |
| Sets, relations, functions | Set identities, equivalence relations, bijections | Proving properties from definitions |
| Proof techniques | Direct proof, contradiction, contrapositive, induction | Choosing the right method |
| Combinatorics | Permutations, combinations, inclusion-exclusion, pigeonhole | Knowing whether order matters |
| Recurrences | Solving linear recurrences, generating functions | Characteristic equations |
| Graph theory | Paths, trees, coloring, Euler and Hamilton circuits | Reasoning about degrees and structure |
Logic: Truth Tables and Equivalences
Logic questions usually ask you to show two statements are equivalent, or that one implies another. A truth table settles it by checking every case; algebraic laws give a shorter, more elegant proof.
The implication p → q is false only when p is true and q is false. It is logically equivalent to its contrapositive ¬q → ¬p, but not to its converse q → p.
| p | q | p → q | ¬q → ¬p | q → p |
|---|---|---|---|---|
| T | T | T | T | T |
| T | F | F | F | T |
| F | T | T | T | F |
| F | F | T | T | T |
The matching columns for p → q and its contrapositive are why proof by contrapositive is valid. When negating quantified statements, swap the quantifier and negate the inside: the negation of "for all x, P(x)" is "there exists x such that not P(x)".
Proof Techniques and When to Use Them
- Direct proof: assume the hypothesis and reason forward to the conclusion. Good for simple statements about integers.
- Contrapositive: prove "if not q, then not p". Useful when the negation of the conclusion is easier to work with.
- Contradiction: assume the statement is false and derive something impossible. The classic example is the irrationality of the square root of 2.
- Induction: prove a base case and that each case implies the next. The standard tool for statements about all natural numbers.
- Counterexample: to disprove a "for all" claim, one specific case where it fails is enough.
Worked example: proof by induction. Claim: for every integer n ≥ 1, 1 + 2 + ... + n = n(n + 1)/2.
Base case: for n = 1 the left side is 1 and the right side is 1 × 2 / 2 = 1. True.
Inductive step: assume the claim holds for some k ≥ 1, so 1 + ... + k = k(k + 1)/2. Then 1 + ... + k + (k + 1) = k(k + 1)/2 + (k + 1) = (k + 1)(k/2 + 1) = (k + 1)(k + 2)/2, which is the claim for n = k + 1.
Conclusion: by the principle of mathematical induction, the formula holds for all n ≥ 1.
Notice the structure: base case, a clearly stated inductive hypothesis, the step where the hypothesis is used, and a closing sentence. Markers look for all four.
Sets, Relations and Functions
Set proofs are proofs from definitions. To show A = B, prove A ⊆ B and B ⊆ A separately: take an arbitrary element of one set and show it belongs to the other.
Relations questions usually ask you to check properties. An equivalence relation is reflexive, symmetric and transitive; a partial order is reflexive, antisymmetric and transitive. Test each property against its definition and give a counterexample for any that fail.
- Injective: f(a) = f(b) implies a = b, so no two inputs share an output.
- Surjective: every element of the codomain is hit by some input.
- Bijective: both at once, which means f has an inverse.
For example, f(x) = 2x from the integers to the integers is injective but not surjective, because no integer maps to 3.
Combinatorics: Counting Without Double-Counting
The first question in any counting problem is whether order matters and whether repetition is allowed. That choice picks the formula.
| Situation | Formula | Example |
|---|---|---|
| Ordered, no repetition | P(n, r) = n! / (n − r)! | Gold, silver, bronze from 10 runners: 10 × 9 × 8 = 720 |
| Unordered, no repetition | C(n, r) = n! / (r!(n − r)!) | A committee of 3 from 10: 120 |
| Ordered, repetition allowed | nr | 4-digit PINs: 104 = 10,000 |
| Unordered, repetition allowed | C(n + r − 1, r) | 3 scoops from 5 flavors: C(7, 3) = 35 |
Inclusion-exclusion handles overlaps: |A ∪ B| = |A| + |B| − |A ∩ B|. The pigeonhole principle says that if n + 1 objects go into n boxes, at least one box holds two or more.
Worked example: inclusion-exclusion. How many integers from 1 to 100 are divisible by 3 or 5? Divisible by 3: ⌊100/3⌋ = 33. By 5: 20. By both, so by 15: ⌊100/15⌋ = 6. Total = 33 + 20 − 6 = 47.
Recurrence Relations
A linear homogeneous recurrence such as an = 5an−1 − 6an−2 is solved through its characteristic equation r2 − 5r + 6 = 0, which has roots 2 and 3. The general solution is an = A·2n + B·3n.
Use the initial conditions to find A and B. With a0 = 1 and a1 = 4: A + B = 1 and 2A + 3B = 4, giving B = 2 and A = −1, so an = 2 · 3n − 2n. Check: a2 = 18 − 4 = 14, and 5 × 4 − 6 × 1 = 14.
Graph Theory Essentials
Graph theory questions reward precise definitions. Know the difference between a walk, a trail, a path and a cycle before you start proving anything.
- Handshake lemma: the sum of all vertex degrees equals twice the number of edges, so every graph has an even number of odd-degree vertices.
- Trees: a tree with n vertices has exactly n − 1 edges, and there is exactly one path between any two vertices.
- Euler circuits: a connected graph has one if and only if every vertex has even degree.
- Planar graphs: for a connected planar graph, Euler's formula gives V − E + F = 2.
Hamilton circuits have no simple degree test like Euler circuits do. To show none exists, you usually need a structural argument rather than a formula.
Common Discrete Maths Mistakes
| Mistake | Fix |
|---|---|
| Proving the converse instead of the statement | Write down exactly what is assumed and what must be shown |
| Induction step that never uses the hypothesis | Point to the line where the hypothesis is applied |
| Proving a "for all" claim with examples | Examples only disprove; give a general argument |
| Using permutations when order does not matter | Ask whether swapping two items gives a new outcome |
| Double-counting overlapping cases | Use inclusion-exclusion |
How STEM Donkey Helps with Discrete Mathematics
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Frequently Asked Questions
Logic, sets, relations, functions, proof techniques, combinatorics, recurrences, probability on finite sets and graph theory.
Try a direct proof first. If the conclusion is a negative statement or hard to reach, try the contrapositive or contradiction. For claims about every natural number, induction is usually the tool.
Ordinary induction assumes the claim for k to prove it for k + 1. Strong induction assumes it for every value up to k, which helps when a case depends on several earlier ones.
Order matters when swapping two chosen items creates a different outcome, such as positions in a race. If it does not, as in choosing a committee, use combinations.
A counterexample can only disprove a universal claim. To prove an existence claim, one example is enough; to prove a universal claim, you need a general argument.
Yes. Ask for LaTeX in your brief and you receive the source alongside a PDF.
Heavily. Logic underpins circuits and programming, induction proves algorithms correct, counting drives complexity analysis and graphs model networks.