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Practice Exam 1
CIS/CSE 607, Spring 2009
Problem 1) Let R be a reflexive binary relation on a set A. Prove that R is transitive if, and only if, R = R ◦ R.
Problem 2) Give an example of a transitive binary relation on a set A for which R 6 = R ◦ R
Problem 3) How many binary relations are there, on a set with 3 elements, that have all three properties: reflexive, antisymmetric and transitive? In other words, how many partial orderings are there on a set with 3 elements?
Problem 4) Consider the set of all permutations of a set A. [Remember that a permutation of A is just a one-to-one and onto function from A to A.] Explain why a permutation of A is a binary relation on A. Now let f be the permutation of the set { 1 , 2 , 3 } defined by the input-output pairs f (1) = 2, f (2) = 3 and f (3) = 1. Give the input-output pairs for f 2 , f 3 and f −^1.
Problem 5) Consider again the set of all permutations of a set A. Let I be the identity permutation: i.e. I(x) = x for all x ∈ A. Let π 0 be one of these permutations. Consider the set of permutations
{... , (π− 0 1 )^2 , π 0 − 1 , I, π 0 , π^20 ,... , πn 0 ,.. .}
For two permutations f and g (possibly, f = g) of A, let f ∼ g if, and only if, there is an integer k such that f ◦ g−^1 = πk 0. Prove that ∼ is an equivalence relation on the set of all permutations of A. [For permutation f of A, f −^1
is the inverse permutation. That is, (f ◦ f −^1 )(x) = (f −^1 ◦ f )(x) = x, for all x ∈ A.]
Problem 6) Let A = { 1 , 2 , 3 }. List all six permutations of A. List the equivalence classes of the ∼ relation defined in problem 5, where π 0 is the permutation f given in problem 4.
Problem 7) Recall from exam 1: Let U be a nonempty set. Let I be the operation defined on the power set of U as follows: for each subset A of U and each subset B of U , let
A I B = (U − A) ∪ B
Let U be a nonempty set (i.e. U 6 = ∅). For any function f with domain U and codomain { 0 , 1 } the preimage f −^1 ({ 1 }) of the element 1 in the codomain is a subset of U. Conversely, given any subset A of U , the characteristic function of A is defined by
χA(x) =
0 if x 6 ∈ A 1 if x ∈ A
These remarks establish a one-to-one correspondence between the power set of U and the set of functions with domain U and codomain { 0 , 1 }. For subsets A and B, the following formulas express the characteristic function of their difference, union, and intersection in terms of the characteristic functions χA and χB. (∗ is multiplication.)
χU −A(x) = 1 − χA(x)
χA∩B (x) = χA(x) ∗ χB (x)
χA⊕B (x) = χA(x) + χB (x) − 2 ∗ χA(x) ∗ χB (x)
χA−B (x) = χA(x) ∗ (1 − χB (x))
Give a similar formula for χAIB in terms of χA and χB.
Problem 11) Recall the Fibonacci numbers: f 0 = 0, f 1 = 1, fn = fn− 2 + fn− 1 , where n ≥ 2. For which values of n is fn mod 3 = 0? Prove your answer by induction.
Problem 12) How many bit strings of length 17 are there that contain exactly 5 0’s and exactly 12 1’s?
Problem 13) Let A and B be finite sets. Assume that the number of elements in A is m and the number of elements in B is n. Also assume m ≥ n. Let g : B −→ A be a one-to-one function. In terms of m and n, how many functions f : A −→ B are there such that the following condition is satisfied: f (g(x)) = x for every x in B? Briefly explain your reasoning.
Problem 14) Let Z be the set of integers, and let
P = {n ∈ Z | 1 ≤ n ≤ 20 and n is odd}
and Q = {n ∈ Z | 1 ≤ n ≤ 20 and n is a multiple of 5}
For any two sets U and V , let
U − V = {u ∈ U | u 6 ∈ V }
and U ∆V = (U
V ) − (U
V ).
List the elements in P ∆Q.
Problem 15) Recall that the product P × Q of two sets P and Q is a set of ordered pairs determined by the following definition:
P × Q = {(p, q) | p ∈ P and q ∈ Q}
Let S, A and B be sets such that S ⊆ A × B. Suppose S = {(0, 0), (1, 1)}. Prove that there are no sets D and E such that D ⊆ A, E ⊆ B, and S = D × E.
Problem 16) A (binary) relation R over a set A is simply a set of pairs of elements drawn from A; that is, R ⊆ A × A. The inverse of a relation R is the relation R−^1 defined by:
R−^1 = {(x, y) | (y, x) ∈ R}
The composition R 1 ◦ R 2 of relations R 1 and R 2 is defined by:
R 1 ◦ R 2 = {(x, z) | ∃y. ( (x, y) ∈ R 1 ∧ (y, z) ∈ R 2 )}.
Rn^ is the result of composing n copies of R. More precisely, Rn^ is defined as follows:
R^1 = R Rn+1^ = R ◦ Rn^ (for n ≥ 1)
A relation R is transitive if it satisfies the following property:
∀x, y, z. [(x, y) ∈ R ∧ (y, z) ∈ R =⇒ (x, z) ∈ R]
Let T and U be relations over the set A = { 1 , 2 , 3 , 4 }, as follows:
T = {(1, 1), (2, 1), (3, 3), (4, 4), (3, 4)} U = {(2, 4), (1, 3), (3, 3), (3, 2)}
Calculate the following relations:
- T −^1
- U ◦ T
- U 3
Problem 17) Is the following statment true?
is read as x is way below y.
The set of all elements {y | y x} is denoted by ↓↓x.
Definition: (compact element) Let D be a dcpo. An element x of D is compact iff x x.
Definition: (continuous domain). A continuous dcpo (also called a contin- uous domain) is a dcpo such that for every x ∈ D, x = t(↓↓x)
Definition: The successor of a set S is the set S ∪ {S}.
(a) Let ω be the set of nonnegative integers. Consider the successor of ω, which we denote by ω′, with the ordering obtained by extending the natural ordering on the nonnegative integers so that x ≤ ω for all elements x in ω′. Show that the resulting system is a continuous dcpo.
(b) Are there any compact elements in ω′? Which element(s) is (are) com- pact? Explain.
One way to get an insight into answering these questions is to determine for yourself what the directed subsets of ω′^ look like.
Problem 21) A trit is an element of the set { 0 , 1 , 2 }. Let us call a string σ of trits a good trit-string iff σ does not have two consecutive 0′s in it anywhere. How many good trit-strings are there of length 6.
