5) (Aug '00) Let A denote an arbitrary set, and let R denote a transitive relation over A,
that is,
R  A  A, and for all x, y, z  A, if (x, y)  R and (y, z)  R then (x, z)  R. Prove
that the composition relation R2 = R o R is transitive.
5) (Nov '00) Let A denote an arbitrary non-empty set, and let L denote the relation
defined over A as follows:
L = {(a, a) | a  A}
Suppose R is a transitive relation over A, that is, R  A  A, and for all x, y, z  A, if
(x, y)  R and (y, z)  R then (x, z)  R. Prove that the relation R  L is transitive.
3) (Dec '00) Let A denote an arbitrary non-empty set, and let R denote a binary relation, R
 A  A.
Answer the following two parts independently of each other:
(a) Suppose R is transitive. Prove that the inverse relation R1 is also transitive,
where R1 is defined as R1 = {(a, b) | (b, a)  R}.
(b) Suppose R   and R is irreflexive (that is, there does not exist any a  A
such that (a, a)  R). Prove that either R is not symmetric or R is not
transitive. (Hint: Use an indirect method.)
4) (Dec '00) Let A denote an arbitrary non-empty set, and let R, S, and T denote binary
relations defined over A, i.e., R  A  A, S  A  A, and T  A  A. Answer the
following two questions independently of each of other:
(a) Prove (R  (S  T))  ((R  S)  (R  T)).
(b) Suppose A = {a, b, c}. Use an example of relations R, S, and T defined over
this A to show that (R  (S  T))  ((R  S)  (R  T)).
4) (Mar '01) Let R be a relation with R  A  A, where |A| = 5. Answer the following
questions, giving justifications for your answers.
a) Give an example of a non-empty relation that is both symmetric and anti-symmetric.
(Let the set A = {a, b, c, d, e}.)
b) How many relations R are symmetric?
4) (May '01) a)
Prove or disprove: If R1  R2 and S1  S2, then R1 S1  R2 S2.
b) Prove or disprove: If R1  R2 and S1  S2, then R1 S1  R2 S2.
6) (May '01) Consider the relation R on the set of integers Z={…2, 1, 0, 1, 2, …}:
R ={(x, y) | x, y A and x  y (mod 5)}.
i) Prove that this relation is an equivalence relation.
ii) How many equivalence classes are induced by this relation on Z?
(Recall that: x  y (mod 5) means that 5 | (xy), i. e. there exists an integer k, such that
xy = 5k For instance, 1  6 (mod 5) or 8  3 (mod 5)).
5) (Dec '01) Let R denote a relation on a set A, i.e. R  AA. Assume that R is
symmetric and transitive.
a) Prove or disprove that R is reflexive.
b) Prove or disprove that RR  R.
3) (May '02) Suppose R, S  A  A are two symmetric relations on a set A. Prove or
disprove each of the following propositions.
a) the relation R  S is symmetric.
b) the relation R  S S R is symmetric
6) (Aug '02)
a) Prove that a relation R A A is transitive if and only if R2  R.
b) Prove or disprove: If a relation RA A is symmetric then R2 is symmetric as well.
4) (Dec '02)
a) Consider the following relation defined over Z  Z: R1 = {(x, y) | xy  1 (mod 10)}.
Using counter examples, show that the relation is NOT reflexive, NOT irreflexive, NOT
anti-symmetric, and NOT transitive. (Note: Z is the set of integers.)
b) Prove that the following relation R2 defined over Z  Z is transitive:
R2 = {(x, y) | x +3y = 4c, for some integer c.}
6) (May '03) Let S be a relation over the positive real numbers (so S  R+ x R+,) defined
as follows:
S = { (a,b)) | a=bn, for some positive rational number n.}
Is S reflexive? symmetric? transitive? Give proof for each of your answers.
4) (Aug '03) The set K contains all three-digit integers from 000 to 999.
Thus, K = { 000, 001, 002, … , 998, 999 } . Let n and m be arbitrary members of K. We
will represent the individual digits of n and m as n = n1n2n3 and m = m1m2m3 .
Let R be a relation on K x K such that:
R = { ( n, m) | n  K, m  K,  n1 + n2 + n3 = m1 + m2 + m3}
a) Is R an equivalence relation? Why or why not?
b)If R is an equivalence relation, how many equivalence classes are there?
3) (Dec '03) Relations
a) Show that a relation R over the integers (RZZ) defined as R={(a,b) | a2=b2
and a,bZ} is an equivalence relation.
b) Is the relation R given in part (a) a function? Briefly, why or why not?
3) (May '04) Relations
a) Let A = {1, 2, 3, 4, 5}. R is a relation defined on A, (so RAxA.) In particular,
R = { (1,1), (1, 2), (1, 4), (1, 5), (2, 1), (2, 2), (2, 4), (3, 3), (4, 1), (5, 1) }. Is R
(i)reflexive? (ii)symmetric? (iii)transitive? Prove your answers.
b) Define R to be a relation over the positive integers as follows:
R = { (a,b) | a = cb, for some integer c}
Prove that R is a partial-ordering relation. (Show that it is reflexive,
anti-symmetric and transitive.)
5) (Aug '04) Given an arbitrary set A, relation R on A, i.e. RAA, satisfies the
following property P: aA bA, |R{(a,b), (b,a)}|  1.
Answer the following questions and justify each one of your answers by a simple
proof or a (counter-)example.
(i)
Can R be symmetric?
(ii)
Is R necessarily anti-symmetric?
(iii) Is R necessarily transitive?
(iv)
Can R be reflexive?
3) (Dec '04) Relations
a) Let A = {1, 2, 3, 4, 5}. R is a relation defined on A, (so RAxA.) In particular, R= {(1,
3), (1, 5), (1, 1), (3, 1), (3, 3), (3, 5), (5, 1), (5, 3), (2, 2), (4, 4)}
Is R (i) reflexive? (ii) symmetric? (iii) transitive? Prove your answers.
b) For natural numbers a and b, define aRb iff a2+b is even. Prove that R defines an
equivalence relation on N.
4) (May '05) Relations
Let A = {1, 2, 3, 4, ..., 100}. Define the function u(x) = the units digit of the positive
integer x. For example u(37) = 7 and u(95) = 5.
(a) Let a relation R over A be defined as follows:
R = { (a, b) | aA  bA  u(a4) = u(b4) }. Prove that R is an equivalence relation.
(b) How many equivalence classes does R have?
(c) How many elements are in each equivalence class of R?
4) (Aug '05) Relations
(a) (10 pts) Let A = {1, 2, 3, 4, 5}. Let R be a binary relation over the set A. (Thus, R 
AxA.) In particular R = { (1, 2), (3, 1), (4, 2), (2, 4), (3, 3), (5, 2) }. Explain why R is not
reflexive, irreflexive, symmetric, anti-symmetric or transitive.
(b) (15 pts) Let R be a binary relation over the set of integers defined as follows:
R = { (a, b) | a - b  0 mod 10 }
Prove that R is an equivalence relation.
3) (Dec '05) Relations
a) Let A = {1, 2, 3, 4, 5} and R be a binary relation over A such that R = { (1,1), (1, 3),
(1,5), (2, 2), (2, 4), (3, 1), (3, 3), (4, 2), (5, 1), (5, 3), (5,5)}. Is R reflexive, irreflexive,
symmetric, anti-symmetric, or transitive? Justify each answer.
b) Let R and S be binary relations over Z. Prove or disprove: if R is transitive and S is
transitive, then R  S is also transitive.
3) (May '06) (PRF) Relations
Given the set A = {2, 3, 4, 8, 9, 12, 18}, define a relation T over A such that
T = {(a, b)| a  A and b  A and ab is a perfect square, i.e., ab = c2 for some integer c}.
Answer the following questions.
a) (5 pts) Draw the directed graph representation of the relation T.
b) (15 pts) Determine, with proof, if the relation T satisfies each of the properties:
reflexive, irreflexive, symmetric, anti-symmetric and transitive.
c) (5 pts) Give the matrix representation of T.
4) (Aug '06) (PRF) Relations
(a) Let S = {1, 2, 3, 4} and let A = SS. Define the following relation R on A:
R = {((a, b), (c, d)) | a + b = c + d }
(i) (7 pts) Show that R is an equivalence relation.
(ii) (6 pts) Find the partition A/R
(b) Suppose R and S are symmetric binary relations on a set A. Must the following
relations be symmetric? Give either proofs or counterexamples to justify your answers.
(i) (6 pts) R  S
(ii) (6 pts) R S
6) (Dec '06) (PRF) Relations
(a) (5 pts) Let A = {1, 2, 3, 4, 5}, let B = {a, b, c, d} and let C = {x,y,z}. Furthermore, let
R be a relation over AxB, S be a relation over BxC and T be a relation over BxC. In
particular, let R = { (1,a), (1, c), (2, b), (2, d), (4,a), (4,d), (5,c)}, let S = {(a,z), (c,x),
(c,y), (b,z) }, and let T = { (a,x), (b,x), (b,y), (c,x), (c,z) }. List the elements in the
following two sets: (R  (S  T)) and ((R  S)  (R  T).
(b) (8 pts) Let R be a binary relation over the set of integers, Z. (Thus, R  ZxZ.) Define
the following terms related to binary relations: reflexive, anti-symmetric and transitive.
(c) (12 pts) R-1 denotes the inverse relation of R. (Hence R-1 = { (a,b) | (b,a)  R }.)
Using the direct proof method, show that if R is transitive, then R-1 is transitive.
5) (May '07) (PRF) Relations
(a) (12 pts) Let V be the set of all voters casting a vote for a presidential candidate in a
single election. Each voter votes for exactly one candidate. Define a relation, R: V 
V, that imposes an equivalence relation on the set of voters. Justify that R is an
equivalence relation.
(b) (3 pts) If there are three candidates (with no write-ins allowed) how many equivalence
relations may be created? (Note: let |V| denote the cardinality of the set of voters, V.)
5) (Aug '07) (PRF) Relations
Let A  {1,2,3} and let S  A  A . Define the following relation R on S:
R  {(( a, b), (c, d )) | a 3 c and b  2 d }.
Note that x  m y if x and y are integers and m | ( x  y ) .
(a) (9 pts) Prove that R is an equivalence relation.
(b) (6 pts) Give the partition ( A  A) / R .
5) (Dec '07) (PRF) Relations
Let S  {79, 7379, 2, 1885, 91761, 513786, 357, 21976, 4298} .
relation R on S:
R  {( a, b) | l (a) 5 l (b) } ,
Define
the
following
where l(a) is the number of digits in a. Note that x  m y if x and y are integers and
m | ( x  y) .
(a) (10 pts) Prove that R is an equivalence relation.
(b) (5 pts) Give the partition S / R .