Question

To determine

a) How many nonzero entries does the matrix representing the relation \(R\) on \({\rm{A}} = \{ 1,2,3, \ldots ,100\} \) consisting of the first \(100\) positive integers have if \(R\) is \(\{ (a,b)\mid a > b\} \).

b) How many nonzero entries does the matrix representing the relation \(R\) on \({\rm{A}} = \{ 1,2,3, \ldots ,100\} \) consisting of the first \(100\) positive integers have if \(R\) is \(\{ (a,b)\mid a \ne b\} \).

c) How many nonzero entries does the matrix representing the relation \(R\) on \({\rm{A}} = \{ 1,2,3, \ldots ,100\} \) consisting of the first \(100\) positive integers have if \(R\) is \(\{ (a,b)\mid a = b + 1\} \)?

d) How many nonzero entries does the matrix representing the relation \(R\) on \({\rm{A}} = \{ 1,2,3, \ldots ,100\} \) consisting of the first \(100\) positive integers have if \(R\) is \(\{ (a,b)\mid a = 1\} \) ?

e) How many nonzero entries does the matrix representing the relation \(R\) on \({\rm{A}} = \{ 1,2,3, \ldots ,100\} \) consisting of the first \(100\) positive integers have if \(R\) is \(\{ (a,b)\mid ab = 1\} \).

Short answer

a) The number of non-zero entries is \(4950\).

b) The number of non-zero entries is \(9900\).

c) The number of non-zero entries is \(99\).

d) The number of non-zero entries is \(100\).

e) The number of non-zero entries is \(1\).

Step-by-step solution

Given data

The relation \({\rm{R}}\) on the set \(A = \{ 1,2,3, \ldots ,100\} \) defined by the ordered pairs \(\{ (a,b)\mid a = 1\} \).

Concept of Matrix

The ordered pair\((i,j)\)belongs to the relation if and only if the\({(i,j)^{th}}\)entry in the matrix is\(1\).

Calculation of non – zero entries for \(\{ (a,b)\mid a > b\} \)

a)

Here the total number of entries in the matrix is \({100^2} = 10,000\).

There is a 1 in the matrix for each pair of distinct positive integers not exceeds 100 , namely in position (a, b) where \(a > b\).

Thus the answer is the number of subsets of size 2 from a set of 100 elements, i.e.,

\(\begin{array}{l}C(100,2) = \frac{{100!}}{{2!(100 - 2)!}}\\C(100,2) = \frac{{100 \times 99 \times 98!}}{{2! \times 98!}}\\C(100,2) = 50 \times 99\\C(100,2) = 4950\end{array}\)

Calculation of non – zero entries for \(\{ (a,b)\mid a \ne b\} \)

b)

There is a \(1\) in the matrix at each position except the 100 positions on the main diagonal. Therefore the answer is \({100^2} - 100 = 9900\).

Calculation of non – zero entries for \(\{ (a,b)\mid a = b + 1\} \)

c)There is a 1 in the matrix at each entry just below the main diagonal (i.e., in positions \((2,1),(3,2) \ldots (100,99).\) Therefore the answer is \(99\).

Calculation of non – zero entries for \(\{ (a,b)\mid a = 1\} \)

d)

The entire first row of this matrix corresponds to \(a = 1\).

Therefore the matrix has 100 nonzero entries.

Calculation of non – zero entries for \(\{ (a,b)\mid ab = 1\} \)

e)

This relation has only the one element \((1,1)\). in it, so the matrix has just one nonzero entry.