Question

Problem 1 Provide a real life or a numerical example for cach of the following two cases: 1) A relation that is transitive but not complete. 2) relation that is (x)mplete but not transitive You need to first specify the universe for each of the erample.)

Answer 1

1)

Let us consider a preference relation of "a shape fitting into another shape".

Let us say we have three elements - {small circle, medium triangle, large triangle}.

Small circle fits into the medium triangle, and medium triangle fits into the large triangle implying that small circle also fits into the large triangle, thus transitivity is satisfied.

But now suppose we have a large circle instead of small one. It is quite possible that it won't fit into large triangle but the medium triangle still fits into both. Hence completeness is not satisfied since we don't have well defined relation between every element. Following diagram illustrates this further -

By: W & L INTERMEDIATE MICROECONOMICS PROFESSOR A. JOSEPH GUSE

2)

Consider the popular game - Rock, Paper, Scissor where Rock beats scissor, scissor beats paper but paper beats rock. Due to the cyclic nature, the game is intransitive. It is complete relation though, since between any two elements of a pair, we always know which one is preferred.