Write a complete argument, then compare it with a worked solution and review checklist. Your written reasoning is not automatically graded.
For b>1 and n>0, log_b n is the unique exponent x with bˣ=n. For n≥1 and b>1, exactly ⌈log_b n⌉ repeated divisions by b are needed to first obtain a value at most 1. Halving specifically uses b=2 and log₂ n.
Name the unit being counted. Sequential blocks add their work. Nested loops multiply only when the inner count is the same on every outer iteration; otherwise sum the varying counts. Include work inside each body.
⌊x⌋; ⌈x⌉
For real x, ⌊x⌋ is the greatest integer ≤x; ⌈x⌉ is the least integer ≥x.
The defining bounds are ⌊x⌋≤x<⌊x⌋+1 and ⌈x⌉−1<x≤⌈x⌉. Floor does not generally mean truncating toward zero.
⌊−2.3⌋=−3 and ⌈−2.3⌉=−2; for integer n, both ⌊n⌋ and ⌈n⌉ equal n.
f∈O(g); f∈Ω(g); f∈Θ(g)
For eventually nonnegative functions f(n) and eventually positive g(n) on integer inputs: O(g) means f(n)≤C g(n) for all n≥n₀; Ω(g) means c g(n)≤f(n) for all n≥n₀; Θ(g) requires both bounds. Suitable constants c,C>0 and a threshold n₀ must exist and be independent of n.
Big O alone is an upper bound, not necessarily a tight order. The common shorthand f=O(g) means membership in a class of bounds, not equality with one function. Ω(g) is unrelated to the sample-space name Ω.
For n≥1, 3n²≤3n²+2n+1≤6n², so 3n²+2n+1∈Θ(n²), using c=3,C=6,n₀=1.
T(n)
T(n) counts a specified kind of operation for inputs of size n. State the operation and whether the model describes a particular case, worst case or average case.
This is an exact count under that model; Θ(n²) describes its growth. It is not automatically a time in seconds or the count of every machine instruction.
If iteration i makes i−1 comparisons, for i=1,…,n, then T(n)=∑_{i=1}^{n}(i−1)=n(n−1)/2. For n=4 there are 6 comparisons.
log_b x=y ⇔ b^y=x
For real b>0, b≠1 and x>0, log_b x is the exponent y for which b^y=x. In growth analysis the base is usually a fixed b>1.
Changing between fixed bases greater than 1 multiplies the logarithm by a positive constant. Actual algorithms may round sizes; include that rounding in an exact operation count.
log₂8=3. For n≥1, n/2^k≤1 exactly when k≥log₂n, so the least nonnegative integer k is ⌈log₂n⌉.
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 §11.5 in All topicsConcept in Epp: §11.5, printed page 813 ↗
The condition is checked before the body; 32 is produced in the fifth body execution.