BMath Bachelor of Mathematics Bachelor of Mathematics: Concepts and Theories 2

Question 1

Which of the following best describes a metric space?

Accepted Answer

A set with a function that measures distance satisfying positivity, symmetry, and the triangle inequality

Answer

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)?

Accepted Answer

f achieves every value between f(a) and f(b)

Answer

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:

Accepted Answer

R is commutative, has unity, and has no zero divisors

Answer

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?

Accepted Answer

For every ε>0 there exists N such that d(xₘ,xₙ)<ε for all m,n>N

Answer

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,

Accepted Answer

dim(V) = rank(T) + nullity(T)

Answer

The rank-nullity theorem equates the dimension of the domain to the sum of the rank (dimension of image) and nullity (dimension of kernel).

BMath Bachelor of Mathematics Practice Test