R is a directed graph and can be represented using Adjacency Matrix of 1000x1000 .
Let's take the 1st one :-
$\{ (a, b) | a ≤ b \} $. The first row will have 1000 1's since 1≤{1,2,.....1000}.
The second row will have 999 entries .
Similarly , the pattern will continue and last row will have only one 1 , since 1000≤1000.
Thus total number of non zero entries = $1+2+3+....1000 = \frac{1000*1001}{2} = 500500$.
2nd One:-
$\{ (a, b) | a = b ± 1 \}$.
Leaving the 1st and the last row , every row will have 2 non zero entries . Why? Since 1=2-1 and 1000=999+1.
Thus , total = $1+1+2*998 = 1996+2 = 1998$.
Third One :-
$\{ (a, b) | a + b = 1000 \}$. Here , each row will have one entry , leaving the last row , since $1000+0$ shall be 1000 , but 0 is not in the set , thus total = $999$.
Fourth One :-
$\{ (a, b) | a + b ≤ 1001 \}$.
Here , first row will have 1000 entries , since 1+1000 = 1001 , and every other addition with 1 will be less than 1001.
The second row will have 999 entries , since 2+1000 = 1002 and 2+999 = 1001 , thus it will have one entry less.
This will continue , and the final row will have just one entry $[1000+1=1001]$.
So total number of entries = $1+2+3+....1000 = \frac{1000*1001}{2} = 500500$.
And the last one ,
$\{ (a, b) | a ≠ 0 \}$.
It will have $1000*1000 = 1000000$ non zero entries
