Question

the binary relation S={} (empty set) on a set A={1,2,3,4}
is
1. neither symmetric nor reflexive
2.symmetric and reflexive
3.transitive and reflexive
4.transitive and symmetric

Answer 1

Reflexive says for all x in A, xSx, which is not true, so it is not reflexive. Symmetric says that if xSy then ySx, and transitive says that if xSy and ySz, then xSz. These are both vacuously true. So the answer is 4.

Answer 2

4

Answer 3

Thanks ... But I didnt get the reflexive part a relation on xSx is null so shouldnt the relation also be reflexive ?

Answer 4

Reflexive says that \(\forall x\ (x,x)\in S\). Since S is empty, that isn't true.

Answer 5

alright thanks !