Mathematics

Problem

Let

n

be a natural number greater than 1. Define the sets

A = \{1,2,3,\ldots, n\}

and

B = \{1,2,3,\ldots, n\}

. Consider the subset

S

A \times B

given by

S = \{ (a,b) \in A \times B \mid a > b \}

. a. (i) Prove that

|S| = \frac{n(n-1)}{2}

. b. (ii) For

n = 50

, calculate

|S|

and

|A \times B|

. c. (iii) For general

n

, express the ratio

|S| : |A \times B|

in its simplest form and state what this ratio approaches as

n

becomes very large.