Discrete Math

Question 19 from section 9.3 is already solved, all these problems have answers on chegg, it just takes me too much time too copy them.
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Homework Assignment #9
Question 19 from section 9.3 is already solved, all these problems have answers on chegg, it just
takes me too much time too copy them.
Section 9.1:
1. List the ordered pairs in the relation R from
A = {0, 1, 2, 3, 4} to B = {0, 1, 2, 3}, where (a, b) ? R
if and only if
a) a = b. b) a + b = 4.
c) a > b. d) a | b.
e) gcd(a, b) = 1. f ) lcm(a, b) = 2.
3. For each of these relations on the set {1, 2, 3, 4}, decide
whether it is reflexive, whether it is symmetric, whether
it is antisymmetric, and whether it is transitive.
a) {(2, 2), (2, 3), (2, 4), (3, 2), (3, 3), (3, 4)}
b) {(1, 1), (1, 2), (2, 1), (2, 2), (3, 3), (4, 4)}
c) {(2, 4), (4, 2)}
d) {(1, 2), (2, 3), (3, 4)}
e) {(1, 1), (2, 2), (3, 3), (4, 4)}
f ) {(1, 3), (1, 4), (2, 3), (2, 4), (3, 1), (3, 4)}
7. Determine whether the relation R on the set of all integers
is reflexive, symmetric, antisymmetric, and/or transitive,
where (x, y) ? R if and only if
a) x = y. b) xy = 1.
c) x = y + 1 or x = y – 1.
d) x = y (mod 7). e) x is a multiple of y.
f ) x and y are both negative or both nonnegative.
g) x = y2. h) x = y2.
23. Use quantifiers to express what it means for a relation to
be asymmetric.
25. How many different relations are there from a set with m
elements to a set with n elements?
2
Section 9.3:
2. Represent each of these relations on {1, 2, 3, 4} with a
matrix (with the elements of this set listed in increasing
order).
a) {(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)}
b) {(1, 1), (1, 4), (2, 2), (3, 3), (4, 1)}
c) {(1, 2), (1, 3), (1, 4), (2, 1), (2, 3), (2, 4), (3, 1), (3, 2),
(3, 4), (4, 1), (4, 2), (4, 3)}
d) {(2, 4), (3, 1), (3, 2), (3, 4)}
19. Draw the directed graphs representing each of the relations
from Exercise 2.
a)
b)
1
2
4
3
3
c)
d)
1
2
4
3
4
1
2
4
3
27. list the ordered pairs in the relations represented
by the directed graph.
Section 9.5:
22. Determine whether the relation with the
directed graph shown is an equivalence relation.
57. Consider the equivalence relation from Example 2,
namely, R = {(x, y) | x – y is an integer}.
a) What is the equivalence class of 1 for this equivalence
relation?
b) What is the equivalence class of 1/2 for this equivalence
relation?

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