Draw the arrow diagram and the matrix representation for each relation.
(b)
The domain for relation R is {1, 2, 3, 4}. R = { (1, 2), (3, 4), (2, 3), (3, 2), (2, 1), (3, 1), (4, 3)
Draw the arrow diagram for each relation.
(d) The domain of relation H is a group of four friends. For x, y in the domain, xHy if y is at
least as tall as x. The table below shows each person in the domain and her height.
Name Height
Angie 5'0"
Bernice 5'3"
Carmen 5'3"
Deirdre 5'5"
(e) The domain of relation H is a group of four friends. For x, y in the domain, xHy if y is taller
than x. The table below shows each person in the domain and her height.
Name
Angie
Bernice
Carmen
Deirdre
Height
5'0"
5'3"
5'3"
5'5"
For each relation, indicate whether the relation is:



reflexive, anti-reflexive, or neither
symmetric, anti-symmetric, or neither
transitive or not transitive
(b)The domain of the relation E is the set of all real numbers. For x, y ∈ R, xEy if x ≤ y.
The players on a football team are all weighed on a scale. The scale rounds the weight of every
player to the nearest pound. The number of pounds read off the scale for each player is called his
measured weight. The domain for each of the following relations below is the set of players on
the team. For each relation, indicate whether the relation is:



reflexive, anti-reflexive, or neither
symmetric, anti-symmetric, or neither
transitive or not transitive
(a) Player x is related to player y if the measured weight of player x is at least the measured
weight of player y. No two players on the team have the same measured weight.
(b) Player x is related to player y if the measured weight of player x is at least the measured
weight of player y. There is at least one pair of players on the team who have the same measured
weight. There is also at least one pair of players on the team who have different measured
weights.
Here are two relations defined on the set {a, b, c, d}:


S = { (a, b), (a, c), (c, d), (c, a) }
R = { (b, c), (c, b), (a, d), (d, b) }
Write each relation as a set of ordered pairs.
(b) R ο S
(d) R ο R
Determine whether each relation is an equivalence relation. Justify your answer. If the relation is
an equivalence relation, then describe the partition defined by the equivalence classes.
(a) The domain is a group of people. Person x is related to person y under relation P if x and y
have a common parent (i.e., x and y have the same biological mother or the same biological
father or both). You can assume that there is at least one pair in the group, x and y, such that xPy.
(b) The domain is a group of people. Person x is related to person y under relation M if x and y
have the same biological mother. You can assume that there is at least one pair in the group, x
and y, such that xMy.
Draw the transitive closure of each graph.