Task :
A binary relation on a set of 7 elements contains exactly 20 pairs.
Could it be :
a) a strict partial order relation?
b) a relation of strict linear order?
In strict linear order, any pair of distinct elements is comparable.
My thinking :
a) There can be a total of $7 \times (7 - 1) / 2 = 21$ pairs of elements. We are given $20$ pairs of elements. Since a strict partial order does not have to be linear, it is very likely that a strict partial order can be created. I can't give any other arguments. Maybe I need to demonstrate some example of strictly partial order on $7$ elements with $20$ pairs to prove the existence of a strictly partial order relation?
b) It seems that there certainly cannot be a strict linear linear order relation, because for that , each pair must be comparable, that is, there must be at least $7 \times (7 - 1) / 2 = 21$ pairs of elements. We have only $20$ pairs, but we need $21$ pairs, one pair is missing, so there is no strict linear order relationship. Is the reasoning correct?