BMath Bachelor of Mathematics Bachelor of Mathematics: Concepts and Theories 4 — Questions and Answers
Question 1: What is the order of the group of integers modulo 12 under addition, ℤ₁₂?
- 12 (Correct answer)
- 6
- 4
- φ(12) = 4
Correct answer: 12
The group ℤ₁₂ consists of the 12 residues {0,1,2,...,11} under addition modulo 12, so its order is 12.
Question 2: Which of the following correctly describes the Big-O notation f(n) = O(g(n))?
- There exist positive constants C and N such that |f(n)| ≤ C·g(n) for all n > N (Correct answer)
- f(n)/g(n) → 1 as n → ∞
- |f(n)| ≥ C·g(n) for some constant C
- f(n) = g(n) for all sufficiently large n
Correct answer: There exist positive constants C and N such that |f(n)| ≤ C·g(n) for all n > N
Big-O describes an asymptotic upper bound: f grows no faster than a constant multiple of g beyond some threshold.
Question 3: In complex analysis, a function is holomorphic if:
- It is complex-differentiable at every point in its domain (Correct answer)
- It can be expressed as a power series with real coefficients
- Its real and imaginary parts are continuous
- It maps the real line to the real line
Correct answer: It is complex-differentiable at every point in its domain
Holomorphic functions are complex-differentiable on an open set, which (remarkably) implies they are infinitely differentiable and analytic there.
Question 4: Which of the following is an example of a non-Euclidean geometry?
- Hyperbolic geometry, where the parallel postulate fails (Correct answer)
- Coordinate geometry using the Cartesian plane
- Projective geometry preserving all angles
- Affine geometry with a distance metric
Correct answer: Hyperbolic geometry, where the parallel postulate fails
Hyperbolic geometry replaces Euclid's parallel postulate with the statement that infinitely many parallel lines can pass through a point outside a given line.
Question 5: The Pigeonhole Principle states that if n+1 objects are placed into n containers, then:
- At least one container holds more than one object (Correct answer)
- Every container holds exactly one object
- At least one container is empty
- No two containers hold the same number of objects
Correct answer: At least one container holds more than one object
The Pigeonhole Principle is a basic combinatorial result: distributing more objects than containers forces repetition.
Question 6: In number theory, Fermat's Little Theorem states that if p is prime and gcd(a, p) = 1, then:
- aᵖ⁻¹ ≡ 1 (mod p) (Correct answer)
- aᵖ ≡ 0 (mod p)
- aᵖ ≡ a (mod p) only when a = 1
- a² ≡ 1 (mod p)
Correct answer: aᵖ⁻¹ ≡ 1 (mod p)
Fermat's Little Theorem: for prime p and integer a not divisible by p, a raised to the (p-1)th power is congruent to 1 mod p.
Question 7: Which of the following is a correct characterization of a compact set in ℝⁿ by the Heine-Borel theorem?
- A subset of ℝⁿ is compact if and only if it is closed and bounded (Correct answer)
- A subset of ℝⁿ is compact if and only if it is open and bounded
- A subset of ℝⁿ is compact if and only if it is closed and connected
- A subset of ℝⁿ is compact if and only if it is bounded
Correct answer: A subset of ℝⁿ is compact if and only if it is closed and bounded
The Heine-Borel theorem characterizes compact subsets of ℝⁿ as precisely the closed and bounded ones.
BMath Bachelor of Mathematics Bachelor of Mathematics: Concepts and Theories 4
What is the order of the group of integers modulo 12 under addition, ℤ₁₂?