MA 2322 Worksheet 4 Hints/Solutions

(1) Suppose |A| = n. How many relations on A are there which are

(a) reflexive? 2n2−n

(b) symmetric? 2n2C(n,2)

(c) anti-symmetric? 2n3C(n,2)

(d) reflexive and symmetric? 2C(n,2)

(e) reflexive and anti-symmetric? 3C(n,2)

(f) symmetric and anti-symmetric? 2n

(g) reflexive, symmetric and anti-symmetric? 1

(2) Define a relation ∼ on the set P (Z) by A ∼ B ⇐⇒ A ∩B 6= ∅. Is the relation ∼(a) reflexive? No. ∅ ∩ ∅ = ∅ so ∅ ∼ ∅ is false.(b) symmetric? Yes. If A ∩B 6= ∅ then B ∩ A 6= ∅.(c) anti-symmetric? No. {1} ∩ {1, 2} 6= ∅ and {1, 2} ∩ {1} 6= ∅, but {1, 2} 6= {1}.(d) transitive? No. {1} ∩ {1, 2} 6= ∅ and {1, 2} ∩ {2} 6= ∅ but {1} ∩ {2} = ∅.

(3) Recall that if f : A → A is a function then the graph of f is the set Gf ={(a, f(a))|a ∈ A} which is a subset of A × A and so Gf defines a relation on A.What property must the associated digraph of such a relation have? A vertex acorresponds to an element of A, and a directed edge from vertex a to vertex bcorresponds to the fact that f(a) = b. Therefore, since f is a function and everyinput has exactly one output, the property is that every vertex is the initial vertexof exactly one edge.

(4) How many different equivalence relations are there on the set A if |A| = 5? Apartition of A can consist of 1, 2, 3, 4 or 5 sets. The number of ways to do this canbe counted by first looking at the corresponding partitions of 5 (see WS1) whichare 5, 4 + 1, 3 + 2, 3 + 1 + 1, 2 + 2 + 1, 2 + 1 + 1 + 1 and 1 + 1 + 1 + 1 + 1, andthen counting how many ways you can distribute the 5 elements into subsets ofthese sizes. We have to be careful when counting the number possible distributions

into the 2+2+1 partition. It is 3C(5,2)2

, not 3C(5, 2). (Why?) You can do this in

1 + C(5, 4) + C(5, 3) + C(5, 3) + 3C(5,2)2

+ C(5, 2) + 1 = 52 ways, so there are thismany partitions and hence this many equivalence realtions.

(5) Let S denote the set of students in a MA2322 class, and define a relation R onS by (a, b) ∈ R if and only if students a and b have the same major. Underwhat conditions is R an equivalence relation? The relation is always reflexive andsymmetric. The transitive property can fail if there are double majors.

(6) Find an example of an equivalence relation on R for which(a) every set in the corresponding partition is finite. aRb ⇐⇒ a = b, because

every equivalence class has one element, because a number is only equal toitself.

(b) every set in the corresponding partition is countably infinite. aRb ⇐⇒ a−b ∈Z. Every equivalence class has the same cardinality as Z.

(c) the corresponding partition includes at least one finite set, at least one count-ably infinite set and at least one uncountable set. The easiest way to do this

1

2

is just to choose the partition first: for example A0 = {0}, A1 = Z− {0}, A2 =R− Z and then define aRb if and only if a and b belong to the same Ai.