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 

1Comment
Arvind Rawat @arvind.rawat
8 Jun 2016 07:07 pm

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 set A.

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 A is nonempty.

The statement "S is 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 "S is 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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