The relation ‘is a sister of’ is

Concept — This item uses the traditional (Copi-style) three-way classification of relations by symmetry, in which every relation falls into exactly one of three mutually exclusive classes. A relation is symmetrical when x R y forces y R x in every pair; asymmetrical when x R y rules out y R x in every pair; and non-symmetrical when neither of those universal rules governs it, so the reversal goes through in some pairs and breaks down in others. Transitivity sits on a separate axis: it asks whether x R y together with y R z always forces x R z.

Application — Reverse ‘is a sister of’ in two different pairs:

1. Anita is a sister of Ravi, and Ravi is a boy. Reversing gives ‘Ravi is a sister of Anita’, which is untrue, because a brother is not a sister. So the reversal can break down.

2. Anita is a sister of Meera, and Meera is a girl. Reversing gives ‘Meera is a sister of Anita’, which is true. So the reversal can also go through.

3. The reversal therefore goes through in some pairs and breaks down in others, so neither universal rule governs the swap.

Cross-check — Measure each of the four classifications against that finding:

• symmetrical demands that the reversal go through in every pair, and the Anita–Ravi pair rules that out.

• asymmetrical demands that the reversal break down in every pair, and the Anita–Meera pair rules that out.

• transitive demands that x be a sister of z whenever x is a sister of y and y is a sister of z. Taking Anita and Meera as x and y and Anita again as z forces ‘Anita is a sister of Anita’, which nobody satisfies, so the chain does not close.

• non-symmetrical demands exactly what the two pairs together show: the reversal sometimes goes through and sometimes does not.

Result — ‘is a sister of’ is a non-symmetrical relation.

Note — Some modern set-theory texts use ‘non-symmetric’ in a looser sense, meaning simply ‘not symmetric’, a wider class that also takes in the asymmetrical relations. ‘Is a sister of’ falls under the non-symmetrical label on either reading, because symmetry fails for it.