Consider the set of A all the people who are living down Italy."x lives in the same house as y" is a relation on the set A.Consider the following properties of a relation on a set:
a)Symmetric b)Reflexive c)Transitive
So which one is true?
- a
- b
- c
- a and b
- all of a,b,c
Please explain how to solve this.Thanks! p.s.:-Please mention any good Ebook or Web Site to refer about "Relation & Function"
Symmetric: $xRy \iff yRx$
Is it fair to say that the statements "$x$ lives in the same house as $y$" and "$y$ lives in the same house as $x$" are equivalent? If so, then the relation is symmetric.
Reflexive: $xRx$
Is it always true that "$x$ lives in the same house as $x$"? If so, then the relation is reflexive.
Transitive: $xRy \wedge yRz \implies xRz$
If you are given the two statements "$x$ lives in the same house as $y$" and "$y$ lives in the same house as $z$", can you conclude "$x$ lives in the same house as $z$"? If so, then the relation is transitive.
There is a possible interpretation that would suggest that (c) is the only one not necessarily true (if a person can live in two houses), but absent of that possibility then all three are true.