The question is, "How many nonzero entries does the matrix representing the relation $R$ on $A = \{1,2,3,...,100\}$ consisting of the first $100$ positive integers have if $R = \{(a, b)|a>b\}$ Well, I know that $|A\times A|=100^2$. I also know that any of the ordered pairs where the first element in the pair is one won't won't be in the relation. There is only one ordered pair where 2 is the first element in the ordered pair, and satisfies the condition to be admitted into the relation, namely, $(2,1)$;in a similar fashion, for three, there are only the ordered-pairs $(3, 1)$ and $(3, 2)$. I know that I am required to use some sort of counting techniques, but I am not sure how to implement them.
I would appreciate your help, thank you!
I have another one, the relation is still on the same set, except the condition for an ordered-pair to be in the relation is different: $\{(a,b)|a=b+1\}$. I rewrote the condition as $a-b=1$, just because it was a little more comprehensive.
I reasoned that, for the first row, there will be all in zeros in it; because if $a=1$ and $b=1$, the difference would be zero, which wouldn't satisfy the condition; furthermore, the b values, from then on, become increasingly larger, resulting in negative number differences. I knew from this that the rest of the ninety-nine rows would have at least one element in the row that was one. But as I went out to the fourth row, things became a little more tricky than I suspected. In the forth row, the first element is a zero, (4, 1) doesn't satisfy the condition; but the third element in the fourth row is a 1, (4,3) satisfies the condition. So, I am having a little trouble seeing the pattern.