Food chains
Task number: 4510
The zoo breeds animals of many interesting or endangered species. Some cannot be housed in a common enclosure because one would eat the other. The garden dormouse, for example, cannot be put in an enclosure with the Eurasian lynx, because the lynx would kill off the dormouse. The western hedgehog would defend these two mammalian predators with its spines, but it is too slow (and small relative to the lynx) to threaten them. The Eurasian eagle-owl hunts not only dormice but hedgehogs as well. The eagle-owl will not dare to attack a lynx, but it will easily fly away. Hedgehogs, dormice, and eagle-owls would soon eliminate most burying beetle Nicrophorus vespillo, which feed on the carcasses of vertebrates of all species, but are too small for the lynx. Of these five species, the beetles are the only ones that commonly feed on carcasses. Also, only owls and rodents commit cannibalism (of those mentioned).
Consider the relation \(aFb\) with the meaning "animals of species \(a\) usually feed on the meat of species \(b\)".
Write this relation by listing the elements and by the table, and also represent it by a graph.
Decide whether it is reflexive, symmetric, antisymmetric, or transitive.
What is the meaning and appearance of the relations \(F^{-1}\) and \(F\circ F\)?
Solution
Let us denote the ground set by the initials \(X=\{d,l,h,e,n\}\).
We can then extract the following pairs from the text: \(F=\{ (l,d),(e,d),(e,h),(h,n),(d,n),(e,n),(n,d),(n,l),(n,h),(n,e),(d,d),(e,e) \}\)
Or a table, where \(\times\) denotes a pair from the relation \(F\): \[\begin{array}{cccccc} a\backslash b & d & l & h & e & n \\ d & \times &&&& \times \\ l & \times &&&& \\ h & &&&& \times \\ e & \times && \times & \times &\times \\ n & \times & \times &\times & \times &\\ \end{array} \]
The relation \(F\) is not reflexive because, for example, \((l,l)\notin F\).
Neither is it symmetric because, for example, \((l,d)\in F\) but \((d,l)\notin F\).
Neither is it antisymmetric, because, for example, \((h,n)\in F\) and \((n,h)\in F\).
It is not even transitive, because e.g. \((d,n)\in F\) and \((n,l)\in F\), but \((d,l)\notin F\).
The relation \(F^{-1}\) has the meaning "to be food of". It corresponds to the reversal of elements in pairs, and also to the reversal of arrows in the graph, or \(F^{-1}=\{ (d,l),(d,e),(h,e),(n,h),(n,d),(n,e),(d,n),(l,n),(h,n),(e,n),(d,d),(e,e) \}\)
Similarly, \(a(F\circ F)b\) has the meaning "\(a\) feeds on an animal that feeds on \(b\)" and corresponds to pairs of consecutive arrows, or: \(F\circ F=\{ (d,d),(d,n),(d,l),(d,h),(d,e), (l,d),(l,n), (h,d),(h,l),(h,h),(h,e), \) \( (e,d),(e,n),(e,l),(e,h),(e,e), (n,d),(n,h),(n,e),(n,n) \} \)
The relation \(F\circ F\) is easier to describe by listing the elements of its complement: \(\{(l,l),(l,e),(l,h),(h,n),(n,l)\}\).
