Write a complete argument, then compare it with a worked solution and review checklist. Your written reasoning is not automatically graded.
Fix a vertex order. Entry M[i,j] is in row i and column j. For a simple graph it records adjacency as 0 or 1; for a loop-free multigraph it can record the number of parallel edges. Matrix multiplication is explained in this lesson, not assumed as a prerequisite.
Two vertices are adjacent if an edge joins them directly. Being reachable by several edges is a different condition. For an undirected graph, reversing an edge’s endpoint names does not create a new edge.
M[a,b]
For R⊆A×B on finite ordered sets, M[a,b]=1 if (a,b)∈R and 0 otherwise. State the row order A and column order B.
An entry represents a particular ordered pair. A multigraph adjacency matrix may count parallel edges and therefore have entries greater than 1; that is a different convention.
For A=(1,2), B=(u,v), R={(1,v),(2,u)}, the matrix is [[0,1],[1,0]].
A loop has the same endpoint twice. Parallel edges are distinct edges with the same endpoints. For an undirected multigraph, retain each edge’s identity and endpoint pair.
A simple undirected graph excludes both loops and parallel edges. The degree convention does not by itself specify how a matrix should record loops; state the matrix convention explicitly.
e₁=AB and e₂=AB are two edges, not one duplicated set entry. A loop e₃=AA contributes two to deg(A).
M_ij
For a finite loop-free undirected multigraph with fixed vertex order, M_ij is the number of edges between vertices i and j. The diagonal is zero and M is symmetric.
For a simple graph entries are 0 or 1. A directed graph uses a row-to-column direction and need not give a symmetric matrix. Loop-diagonal conventions must be stated separately.
Two vertices (A,B) joined by two parallel edges have M=[[0,2],[2,0]].
(M²)_ij=∑_k M_ik M_kj
For the adjacency matrix of a finite simple graph, (M²)_ij counts walks of length two from i to j, summed over possible intermediate vertices k.
These are walks, not necessarily paths: returning to a previous vertex is allowed. More generally (M^r)_ij counts length-r walks for nonnegative integer r; M⁰ is the identity matrix.
In the path 1—2—3, (M²)_13=1 from 1—2—3, and (M²)_11=1 from 1—2—1.
M_H=PᵀM_GP
For a vertex bijection f from G to H, put P_uv=1 when f(u)=v and 0 otherwise, with rows in G order and columns in H order. For the same adjacency convention, M_H=PᵀM_GP expresses preserved adjacency.
Pᵀ swaps P’s row and column indices. Right multiplication reorders columns and left multiplication by Pᵀ reorders rows. Reversing the convention for P changes the displayed identity; the matrix does not name an edge bijection.
If G order is (1,2,3), H order is (p,q,r), and f(1)=r,f(2)=p,f(3)=q, then P=[[0,0,1],[1,0,0],[0,1,0]].
Take arbitrary odd integers m and n.
Reason: Their representations follow from the definition of odd.
This is the goal, not something already known.
Reason: An existential conclusion requires constructing a witness.
Expand before introducing s.
Reason: Substitution and the distributive law.
The constructed integer s proves the product odd.
Reason: Closure of integers and the definition of odd; the goal was not assumed.
When to choose: Choose it when definitions turn the hypothesis into usable equations or properties.
For every x in D: P(x) implies Q(x). Given: an arbitrary x in D satisfying P. Goal: Q for that same x.
Start: Introduce the objects and their domains; assume only the stated hypotheses. Expand the relevant definition.
Finish: State Q and explain why arbitrariness gives the universal claim. Never use Q as a premise.
Take arbitrary positive integers a and b with a dividing b.
Reason: These are the hypotheses, not conclusions.
We must establish this inequality.
Reason: Writing a goal does not establish it.
Introduce an integer q witnessing divisibility.
Reason: Definition of a dividing b; a is nonzero.
The quotient is positive.
Reason: Both b and a are positive; division by a is allowed.
A positive integer cannot lie strictly between 0 and 1.
Reason: This uses q being an integer, not positivity alone.
Replace aq by b to obtain the goal.
Reason: Multiplication by a positive a preserves the inequality; b=aq was established above.
When to choose: Use it to refute a universal claim. A failed proof attempt alone does not refute anything.
To refute every x in D satisfying P also satisfies Q, find one x in D with P true and Q false.
Start: Write the exact claim and choose a permitted test object. Verify its hypotheses.
Finish: Show explicitly which conclusion fails; one valid counterexample refutes the universal claim.
Test the claim that every integer greater than 1 is prime.
Reason: A universal statement can be disproved by one permitted instance.
The chosen value satisfies the domain and hypothesis.
Reason: A counterexample must meet every hypothesis.
Nine has a positive divisor other than 1 and itself, so it is not prime.
Reason: Definition of prime. Composite means nonprime only within integers greater than 1.
When to choose: Choose cases when a definition or remainder changes the calculation. First show the cases cover every allowed object.
Given P, cover all possibilities C₁,…,Cₖ and prove Q in each.
Start: Derive the case split from a definition or theorem; do not merely list convenient examples.
Finish: Conclude Q because every allowed object belongs to a handled case.
Let n be an arbitrary odd integer.
Reason: The hypothesis fixes the domain but does not yet provide the required m.
We need to construct an integer m.
Reason: The conclusion is existential: both the equation and integrality must be shown.
Only r=1 or r=3 is possible for odd n.
Reason: Quotient–remainder theorem 4.5.1; residues 0 and 2 make n even. This proves completeness.
Case 1: set m=2q²+q.
Reason: Expand the square and factor 8. Sums and products of integers are integers.
Case 2: set m=2q²+3q+1.
Reason: Expansion, 9=8+1, then factor 8; the constructed m is an integer.
Both possible cases give the required integer; it may depend on the case.
Reason: Exhaustive cases establish the original claim for arbitrary odd n.
When to choose: Use it when negating the conclusion gives a more useful starting definition.
To prove P⇒Q, prove ¬Q⇒¬P. Given ¬Q; goal ¬P.
Start: Write the contrapositive and introduce an arbitrary object satisfying ¬Q.
Finish: Establish ¬P, then invoke equivalence with the original implication.
Prove the statement for every integer n.
Reason: This is Proposition 4.7.4.
Instead prove: if n is odd, then n² is odd.
Reason: P⇒Q is equivalent to ¬Q⇒¬P. For integers, not even means odd. We do not assume n² even here.
Introduce an integer k.
Reason: Definition of odd.
The square is odd.
Reason: Algebra and closure of integers show 2k²+2k is an integer; then use the definition of odd.
The original implication follows from the proved contrapositive.
Reason: Logical equivalence of a conditional and its contrapositive.
When to choose: Use it when the negation creates incompatible established facts.
Assume the hypotheses and the negation of the desired conclusion; derive a contradiction.
Start: State exactly what is temporarily assumed for contradiction.
Finish: Name the conflicting statements and discharge the assumption.
Prove the statement for every integer n.
Reason: This is Proposition 4.7.4.
Assume a counterexample: n² is even but n is not even.
Reason: Negating the universal conditional gives one integer satisfying the hypothesis and negating the conclusion. Every integer is either even or odd.
Introduce an integer k.
Reason: Definition of odd.
The square is odd.
Reason: Algebra and closure of integers show 2k²+2k is an integer; then use the definition of odd.
This is impossible; the assumed counterexample cannot exist.
Reason: No integer is both even and odd (Theorem 4.7.2). Discharging the contradictory assumption proves the original claim.
When to choose: Use it for a claim indexed by every integer n from a starting value, when the next case relates to the previous one.
Prove P(n₀). For arbitrary k≥n₀, assume P(k) and derive P(k+1).
Start: State the indexed property and starting value; check the base without using the induction hypothesis.
Finish: Invoke induction only after both the base and the implication are established.
Prove the sum formula for every positive integer n.
Reason: The domain and indexed property specify what induction must cover.
The first case holds.
Reason: Direct evaluation establishes the base.
Fix an arbitrary integer k and assume only its case.
Reason: This temporary induction hypothesis is permitted to prove the implication P(k)⇒P(k+1).
Separate the next term and replace the old sum by k².
Reason: This is the exact point where the induction hypothesis is used.
This is the required case k+1.
Reason: Algebra proves the implication. The base and induction principle now give all integers n≥1.
When to choose: Use it to identify precisely what would make a quantified claim false, especially before a counterexample or contradiction.
Negate one outer quantifier at a time, preserving variable order, domains and the scope of the predicate.
Start: Mark the scope of each quantifier; move the negation inward one rule at a time.
Finish: Read the final statement and check that it describes failure of the original claim. Equivalence does not by itself prove either statement true.
Rewrite the negation without changing its meaning.
Reason: The domains D and E remain fixed throughout.
There is an x for which the inner existential statement fails.
Reason: Negation of a universal quantifier: ¬∀x R(x) ⇔ ∃x ¬R(x).
For that same x, every y fails P.
Reason: Negation of an existential quantifier: ¬∃y P(x,y) ⇔ ∀y ¬P(x,y). The order ∃x∀y is preserved.
Both are statements established by proof. The name often signals how a text organizes or emphasizes a result, not a different degree of truth.
A proved auxiliary result used in another argument. For example, −|r|≤r≤|r| for real r supports later bounds. If r≥0, |r|=r and −r≤r; if r<0, |r|=−r and r≤−r. These cases cover every real r.
Lemma 4.5.4, printed p. 207 / PDF 231
A result derived from an established theorem. If r is rational, r+r is rational by closure of rational numbers under addition (Theorem 4.3.2); because 2r=r+r, its double is rational.
Example 4.3.4, printed p. 187 / PDF 211; printed box: Corollary 4.2.3
With Epp’s definition, d divides 0 for every nonzero integer d: 0=d·0. Do not omit d≠0.
Prime and composite classifications here concern integers n>1. “Not prime” alone does not make 0, 1 or a negative integer composite.
Enter an integer result. Put any explanation in the reasoning field below.
Check answer checks the result above. Your reasoning is saved, not automatically graded; compare it with the walkthrough.
Back to §10.2 in All topicsConcept in Epp: §10.2, printed page 698 ↗
The zero diagonal does not count as a neighbor.