##### binary relations

2. The binary relation S = ф (empty set) on set A = {1, 2, 3} is

(a) neither reflexive nor symmetric (b) symmetric and reflexive (c) transitive and reflexive (d) transitive and symmetric

(a) neither reflexive nor symmetric (b) symmetric and reflexive (c) transitive and reflexive (d) transitive and symmetric

Whether the empty relation is reflexive or not depends on the set on which you are defining this relation -- you can define the empty relation on any setA.

The statement "S is reflexive" says: for each x∈A, we have (x,x)∈S. This is vacuously true if A=∅(empty set), and it is false if Ais nonempty.

The statement "Sis symmetric" says: if (x,y)∈S then (y,x)∈S. This is vacuously true, since (x,y)∉S for all x,y∈A.

The statement "Sis transitive" says: if (x,y)∈S and (y,z)∈S then (x,z)∈S. Similarly to the above, this is vacuously true.

To summarize, S is an equivalence relation if and only if it is defined on the empty set. It fails to be reflexive if it is defined on a nonempty set.

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