BMath Bachelor of Mathematics Number Theory and Discrete Mathematics 2

Question 1

How many subsets does a set with n elements have?

Accepted Answer

2ⁿ

Answer

Each element is either included or excluded from a subset, giving 2 choices per element and 2ⁿ total subsets.

Question 2

The principle of inclusion-exclusion for two sets A and B states |A ∪ B| equals:

Accepted Answer

|A| + |B| − |A ∩ B|

Answer

Inclusion-exclusion corrects for double-counting: |A ∪ B| = |A| + |B| − |A ∩ B|.

Question 3

A graph is said to be connected if:

Accepted Answer

There exists a path between every pair of vertices

Answer

Connectivity means any two vertices can be linked by a path, so the graph forms one piece rather than isolated components.

Question 4

The number of ways to arrange n distinct objects in a row is:

Accepted Answer

n!

Answer

There are n choices for the first position, n−1 for the second, and so on, giving n! = n × (n−1) × … × 1 total permutations.

Question 5

Euler's formula for connected planar graphs states V − E + F equals:

Accepted Answer

2

Answer

Euler's formula V − E + F = 2 relates vertices V, edges E, and faces F (including the outer face) of any connected planar graph.

BMath Bachelor of Mathematics Practice Test