Download Discrete mathematics I. practice

Discrete mathematics I. practice - Relations and functions
Emil Vatai
February 21, 2017
1. Find the R∩S relation, if R is the m divides n relation
on N, and S is the n = m + 6 relation on Z.
9. On the set N × N let us define a relation R the following way: (m1 , n1 )R(m2 , n2 ), if m1 ≤ m2 and n1 ≤ n2 .
Show that R is a partial ordering.
2. Let X = {a, b, c}. Determine the number of all binary relations on X. Look for examples and counterexamples for properties of relations.
10. Is the following relation a function? R ⊂ A×A, where
A is the set of lines in the plane; aRb, if the lines a and
b form an angle of 60◦ : Examine the properties of this
relation (are they: reflexive, transitive, symmetric?).
3. Let R ⊆ N × N such a relation, that nRm (n, m ∈ N)
is true, if the number of common divisors of n and m
is even. Examine the properties of R.
11. Let A be the set of equilateral triangles, which have
a predefined height m > 0 associated with the base
of the triangle, B = {b : b > 0, b ∈ R}. Let us define
the relation R ⊆ A × B as follows: aRb, a ∈ A, b ∈ B,
if the area of triangle a equals b. Show that R is a
function, and examine its properties (is it injective,
surjective, bijective).
4. Look for relations which are
•
•
•
•
•
•
reflexive, but not transitive;
anti-symmetric and reflexive;
anti-symmetric and not transitive;
not reflexive, not transitive;
not transitive, but trichotomous;
“none” (not reflexive, not transitive, not symmetric, not anti-symmetric, not trichotomous).
12. Determine the domain, range of the following relations, decide if they and their inverses are functions
or not:
• {(x, y) ∈ R2 : a < x < b, x < y < 2x}, where
a, b ∈ R are given numbers;
5. Let us define the following relation on Z and examine
its properties:
• {(x, y) ∈ R2 : y(1 − x2 ) = x − 1},
• xR1 y, if x2 + y 2 is divisible by 2;
• xR2 y, if x2 − y 2 is divisible by 2.
• {(x, y) ∈ R2 : y = (x − 1)/(1 − x2 )};
• {(x, y) ∈ R2 : |x| + |y| ≤ 1};
6. Given a set X prove that ∼ is an equivalence relation!
What are the classes defined by ∼?
•
•
•
•
•
•
•
•
X
X
X
X
X
X
X
X
• {(x, y) ∈ R2 : x2 = 1 + y 2 , y > 0};
• {(x, y) ∈ R2 : x2 + y 2 − 2y < 0}.
= C, z ∼ w, if |z| = |w|;
= C, z ∼ w, if z/|z| = w/|w|;
= C, z ∼ w, if z/w = ±1;
= C, z ∼ w, if z/w ∈ {±1, ±i};
= C, z ∼ w, if (z/w)n = 1;
= Z15 , x ∼ x, if 5(x − y) ≡ 0 mod 15;
= N × N, (p, q) ∼ (r, s), if p + s = r + q;
= Z × N+ , (p, q) ∼ (r, s), if pr = sq.
13. Consider the function f (x) = x/(1 − x). It cannot
be defined for the entire number line. Which is the
widest subset of real numbers, for which f can be
defined? Is this function injective/surjective/bijective
on this subset of real numbers? What is the widest
subset of R, for which every composition f, f ◦ f, f ◦
f ◦ f, . . . can be defined? What can we say about the
connection of f ◦ f ◦ f and f ?
14. For the given relations on R determine dmn(R),
rng(R). For A = {0, 1, 2} determine the image of
A i.e. R(A), preimage R−1 (A), restriction R |A :
7. Write a program which decides if a relation is reflexive (symmetric, anti-symmetric, transitive) or not.
Count the number of equivalence relations, and the
number of partial orderings defined on a four element
set using the program.
(a) R = {(x, y) ∈ Z3 × Z3 : y 2 = x},
(b) R = {(x, y) ∈ Z4 × Z4 : y 2 = x2 },
8. Adam and Bob are playing the following game: First
Adam selects a subset of the X = {1, 2, 3} set, then
Bob selects a subset of X, than again Adam and so
on. But they must never select a subset which was
a subset of a previously selected (sub)set. The loser
is the one who makes the last move, which has to
be selecting the entire X set. A possible game play:
A : {1}, B : {2}, A : {1, 3}, B : {2, 3}, A : {1, 2},
B : {1, 2, 3}, and Adam wins. What is the winning
strategy?
(c) R = {(x, y) ∈ Z5 × Z5 : y 2 = x3 + x + 1},
(d) R = {(x, y) ∈ Z7 × Z7 : y 2 = x3 + 2}.
15. Determine S ◦ R and R ◦ S, if
• R = {(x, y) ∈ Z3 × Z3 : y 2 = x} and S =
{(x, y) ∈ Z3 × Z3 : y = 2x};
• R = {(x, y) ∈ Z4 × Z4 : y 2 = x2 } and S =
{(x, y) ∈ Z4 × Z4 : y = 2x};
1
• R = {(x, y) ∈ Z5 × Z5 : y 2 = x3 + x + 1} and
S = {(x, y) ∈ Z5 × Z5 : y 2 = x3 + 1};
• R = {(x, y) ∈ Z6 × Z6 : xy = 0} and S =
{(x, y) ∈ Z6 × Z6 : x2 + y 2 = 0};
• R = {(x, y) ∈ Z7 × Z7 : xy = 1} and S =
{(x, y) ∈ Z7 × Z7 : x2 + y 2 = 1};
• R = {(x, y) ∈ Z8 × Z8 : x2 y 2 = 0} and S =
{(x, y) ∈ Z8 × Z8 : x2 + y 2 = 0};
16. Determine the domain, range, of the following relations, determine if they are functions and if their inverses are functions:
• R = {(x, y) ∈ Z5 × Z5 : x · y = 1};
• R = {(x, y) ∈ Z6 × Z6 : x · y = 1};
• R = {(x, y) ∈ Z5 × Z5 : x2 + y 2 = 1};
• R = {(x, y) ∈ Z5 × Z5 : x4 − x = y};
• R = {(x, y) ∈ Z7 × Z12 : x4 − x = y};
• R = {(x, y) ∈ Z8 × Z3 : x4 − x = y};
• R = {(x, y) ∈ Z8 × Z7 : x6 = y}.
Solutions
2