BMath Bachelor of Mathematics Bachelor of Mathematics: Concepts and Theories 2 — Questions and Answers
Question 1: Which of the following best describes a metric space?
- A set with a function that measures distance satisfying positivity, symmetry, and the triangle inequality (Correct answer)
- A set where every subset is open
- A vector space equipped with an inner product
- A topological space where every sequence converges
Correct answer: A set with a function that measures distance satisfying positivity, symmetry, and the triangle inequality
A metric space is a set X with a distance function d(x,y) satisfying non-negativity, d(x,y)=0 iff x=y, symmetry, and the triangle inequality.
Question 2: What does the Intermediate Value Theorem guarantee for a continuous function f on [a, b] with f(a) < 0 < f(b)?
- f achieves every value between f(a) and f(b) (Correct answer)
- f is differentiable on (a, b)
- f has a unique zero in (a, b)
- f is bounded on [a, b]
Correct answer: f achieves every value between f(a) and f(b)
The IVT states that a continuous function on a closed interval takes every intermediate value between its endpoint values.
Question 3: In ring theory, a ring R is called an integral domain if:
- R is commutative, has unity, and has no zero divisors (Correct answer)
- Every element of R has a multiplicative inverse
- R is a field with characteristic 0
- R has exactly one maximal ideal
Correct answer: R is commutative, has unity, and has no zero divisors
An integral domain is a commutative ring with unity where the product of any two nonzero elements is nonzero.
Question 4: Which statement correctly defines a Cauchy sequence in a metric space?
- For every ε>0 there exists N such that d(xₘ,xₙ)<ε for all m,n>N (Correct answer)
- The sequence is bounded and monotone
- Every subsequence converges to the same limit
- The sequence satisfies |xₙ₊₁ - xₙ| < 1/n for all n
Correct answer: For every ε>0 there exists N such that d(xₘ,xₙ)<ε for all m,n>N
A Cauchy sequence requires that terms become arbitrarily close to each other (not just to a fixed limit) as the indices grow.
Question 5: The rank-nullity theorem states that for a linear map T: V → W,
- dim(V) = rank(T) + nullity(T) (Correct answer)
- rank(T) = dim(W) - nullity(T)
- nullity(T) = dim(V) - dim(W)
- dim(V) + rank(T) = dim(W)
Correct answer: dim(V) = rank(T) + nullity(T)
The rank-nullity theorem equates the dimension of the domain to the sum of the rank (dimension of image) and nullity (dimension of kernel).
Question 6: In probability theory, two events A and B are independent if and only if:
- P(A ∩ B) = P(A) · P(B) (Correct answer)
- P(A | B) = P(B)
- P(A ∪ B) = P(A) + P(B)
- P(A ∩ B) = 0
Correct answer: P(A ∩ B) = P(A) · P(B)
Independence means that the occurrence of one event does not affect the probability of the other, formalized as P(A∩B) = P(A)P(B).
Question 7: Which property distinguishes a Hilbert space from a general Banach space?
- A Hilbert space has an inner product whose induced norm makes it complete (Correct answer)
- A Hilbert space is finite-dimensional
- A Hilbert space contains no non-convergent Cauchy sequences
- A Hilbert space has a countable basis
Correct answer: A Hilbert space has an inner product whose induced norm makes it complete
A Hilbert space is a complete inner product space; all Hilbert spaces are Banach spaces but not vice versa since the Banach space norm need not come from an inner product.
BMath Bachelor of Mathematics Bachelor of Mathematics: Concepts and Theories 2
Which of the following best describes a metric space?