Hi, I just want to make sure I am doing this right:
Construct a relation on the set {a, b, c, d} that is
a) reflexive, symmetric, but not transitive.
b) irrreflexive, symmetric, and transitive.
c) irreflexive, antisymmetric, and not transitive.
d) reflexive, neither symmetric nor anti symmetric, and
transitive.
e) neither reflexive, irreflexive, symmetric, antisymmetric,
nor transitive.
What I got is
a) a and b have a common grandparent
b) ab ≥1
c) a = b + 1
d) a mult. of b
e) not sure
and are there more examples other than this?
Thank you!
It looks like you've made some progress in constructing relations that satisfy the given conditions. Let's analyze your answers and discuss if there are other instances:
a) For a relation to be reflexive, symmetric, but not transitive, one possible example is to have pairs (a, b), (b, a), (b, c), (c, b), (a, c), and (c, a) in the relation. This means that each element is related to itself, and there is a mutual relationship between every pair of distinct elements, but the transitive condition is not satisfied. Your example of having a and b share a common grandparent fits this criterion.
b) In an irreflexive, symmetric, and transitive relation, no element should be related to itself (irreflexive), and there should be a mutual relationship between every pair of distinct elements (symmetric and transitive). Your example of ab ≥ 1 is irreflexive since a*a or b*b is less than 1, which violates the irreflexive condition.
c) An irreflexive, antisymmetric, and not transitive relation means no element is related to itself (irreflexive), for any pair (a, b) in the relation, if a is related to b, then b should not be related to a (antisymmetric), and there is at least one pair (a, b), (b, c) in the relation such that (a, c) is not in the relation (not transitive). Your example of a = b + 1 satisfies these conditions.
d) For a relation to be reflexive, neither symmetric nor antisymmetric, and transitive, one possible example is to have pairs (a, a), (b, b), and (a, b) in the relation. This satisfies the reflexive and transitive conditions, but it is neither symmetric nor antisymmetric since both (a, b) and (b, a) are in the relation.
e) A relation that is neither reflexive, irreflexive, symmetric, antisymmetric, nor transitive means it does not satisfy any of the given conditions. In other words, there is no particular relationship or pattern among the elements of the set. As such, any combination of pairs that does not meet the conditions can be considered an example.
It's important to note that there can be multiple correct answers for each case, as long as the relation satisfies the specified conditions. Feel free to come up with other examples as well!