Mathematics

Problem

Let

A=\{1,2,3,\dots ,20\}

. A relation

R

A

is defined by

R=\{(a,b)\in A\times A\mid a \text{ is a multiple of } b\}.

Using this definition, answer the following questions without listing every ordered pair: a. (i) Show that the number of ordered pairs in

R

equals

\displaystyle \sum_{b=1}^{20}\left\lfloor\frac{20}{b}\right\rfloor

. Hence find the exact value of

|R|

. b. (ii) Determine the domain and the range of

R

. c. (iii) Prove that

R

is antisymmetric, giving a clear justification for every step of your proof.