Let H be the half-sibling relation on the set of all people in the world. Is H reflexive? Is H symmetric? Is H antisymmetric? Is H transitive?
Can anyone answer the above questions with reasoning just to understand the way of thinking?
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Reflexive -- can you be your own half sibling?
Symmetry -- If Tom is your half-brother are you Tom's half-sibling?
Transativity -- This is a little trickier. Suppose Tom is your half-brother -- you have the same father and different mothers, and Suzy is Tom's half-sister, Tom and Suzy have the same mother and different fathers, are you related to Suzy?
Anti-symmetric -- this is a more difficult concept. If a member is related to a different member $R(a,b), a\ne b$ then the symmetric $R(b,a)$ is never true. An example of an anti-symmetric relation would be parent-child.