Question

Let N be the "direct neighbor" relation defined on the set of all people: if x and y are persons, xNy iff x and y live directly next to each other, such as in two houses or apartments that are directly next to each other. If there is at least one unit between them, then they don't live directly next to teach other. Check all properties that this relation has.

Symmetric

Anti-symmetric

Reflexive

Transitive

Answer 1

The relation is Symmetric

Explanation:

• Symmetry: If aNb means a and be are directly next to each other

aNb then dist(a,b)=1, dist(b,a)=1 then bNa

The relation is symetric.

• Anti-symmetry:If aNb and bNa then a=b. Since the symetry condition is fullfiled, it shows that a and b can be different. The relation is not anti-symmetric

• Reflexive: is aNa? the distance between to persons must be 1.

Dist(a,a)=0 then aNa is not fullfiled and the relation is not reflexive.

• Transitive: If aNb and bNc then aNc. There is a unit between a and c, so the condition is not satisfied. The relation is not transitive.