Use quantified O, Ω and Θ definitions and count operations.
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Try
An algorithm examining every ordered pair of n entries performs n² pair examinations. Another making one pass performs n. A graph of measured seconds may be noisy, but the operation counts reveal a mathematical growth distinction.
Predict
Predict the meaning of an upper bound. For nonnegative functions eventually, f=O(g) means constants C>0 and n₀ exist with f(n)≤Cg(n) for all n≥n₀. Ω supplies a lower bound; Θ supplies both. The constants cannot change with n.
Experiment
Compare n, n log₂n and n² at n=8,16,32 using exact values for these powers of two. Identify which ratios stay constant and which grow. This table suggests a conjecture; use quantified inequalities to establish an asymptotic claim.
Understand
Worked bound: for n≥1, 3n²≤3n²+2n+7≤12n². Thus the expression is Θ(n²) with explicit constants 3 and 12. To prove n² is not O(n), suppose n²≤Cn eventually. Dividing by positive n gives n≤C, contradicted by choosing a larger integer.
Loop structure must be counted before classified. A triangular nested loop with inner counts 1,2,…,n performs n(n+1)/2 operations, still Θ(n²), not exactly n². A doubling loop from 1 takes ceil(log₂n) doublings to reach at least n for n≥1.
Invalid interpretation: O(n²) says the program takes n² seconds. It specifies an eventual bound on a declared cost function, not hardware timing or an exact formula.
Apply
Name the input size and counted operation, and distinguish worst-case from a specific trace. An O(n³) bound on n² is true but loose. A claim of Θ requires both sides, and testing finitely many inputs cannot establish an eventual universal statement.
Continue
Depth: recurrence methods and amortized analysis explain algorithms whose local costs vary. Randomized expected costs need a stated probability model. Computational complexity theory compares problem families, not only a few implementations on one machine.
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A loop examines every ordered pair of distinct indices from an array of length 9. How many pair examinations?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
For input n=32, a loop doubles k from 1 while k<n. How many doublings?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
The statement f(n)=Θ(n²) means:
Which is a valid but not tight upper bound for n² as n→∞?
Prove 3n²+2n+7=Θ(n²) for positive integer n by giving explicit constants.
Structure your argument and preserve revisions
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Prove n² is not O(n) on the positive integers.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
