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CS123
Spring 2010
Solution to Homework 3
Problem 1 (20 points)
For each of the following 4 relations on the set L of all living people on January 1, 2010, state
(no proof necessary) if the relation is reﬂexive, symmetric, antisymmetric, transitive, equivalence
relation, and/or partial order. In the case of equivalence relations, explain what every equivalence
class signiﬁes. In the case of partial orders, indicate if the set L has a minimum, a maximum,
minimals, and/or maximals, and explain what minimum, maximum, minimal and maximal mean
in that context.
a) x R y if x is taller than y or x is the same person as y.
Solution:
This relation is reﬂexive, not symmetric, antisymmetric, transitive and is a partial order.
In the case that we have only one tallest (shortest) person in the given set L, this person is the
minimum (maximum). The tallest person is also the minimal and the shortest person is also the
maximal. If there are more than one person who tie for the tallest (shortest), then the set L does
not have a minimum (maximum), but does have minimals (maximals), which are these sets of
people.
b) x R y if x and y have the same ﬁrst name.
Solution:
This relation is reﬂexive, symmetric, not antisymmetric (practically speaking), transi
tive, and is an equivalent relation. The equivalence classes are the sets of people having the same
ﬁrst name.
c) x R y if x graduated from the same university as y.
Solution:
This relation is not reﬂexive, symmetric, not antisymmetric (practically speaking),
transitive, and is neither an equivalence relation nor a partial order.
d) x R y if the annual income of x is less than that of y or x is the same person as y.
Solution:
This relation is reﬂexive, not symmetric, antisymmetric, transitive and is a partial order.
In the case that we have only one person who has the most (least) annual income in the given set
L, this person is the maximum (minimum). The person with the most annual income is also the
maximal; The person with the least annual income is also the minimal. If there is a tie for the
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 Spring '09

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