(b) Let P = {...,R_2, R1, Ro, R₁, R₂,...} where R₁ = {x | x € R, [x] n} for any integer n. 2 (i) Draw the real number line from 5 to 5, and indicate the visible sets in P. (ii) Graph the associated binary relation ~p. You only need to graph from -5 to 5. (c) Let P = {E, O} where E = {x | x € R, [x] is even} and O = {x | x € R, [2] is odd}. (i) Draw the real number line from 5 to 5, and indicate the sets E, O EP (a good way to do this is to color the intervals belonging to E and O with different colors). (ii) Graph the associated binary relation ~p. You only need to graph from -5 to 5.
(b) Let P = {...,R_2, R1, Ro, R₁, R₂,...} where R₁ = {x | x € R, [x] n} for any integer n. 2 (i) Draw the real number line from 5 to 5, and indicate the visible sets in P. (ii) Graph the associated binary relation ~p. You only need to graph from -5 to 5. (c) Let P = {E, O} where E = {x | x € R, [x] is even} and O = {x | x € R, [2] is odd}. (i) Draw the real number line from 5 to 5, and indicate the sets E, O EP (a good way to do this is to color the intervals belonging to E and O with different colors). (ii) Graph the associated binary relation ~p. You only need to graph from -5 to 5.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.7: Distinguishable Permutations And Combinations
Problem 29E
Related questions
Question
Please do Exercise 17.2.6 part B and C
Transcribed Image Text:What do partitions have to do with relations? We will illustrate with
the following example.
Let A = {1,2,3,4,5,6} and partition these six numbers into evens and
odds. Then we would have two subsets each with three elements. Suppose
we use a six-sided die to determine a random outcome: where if we get an
even number we win a dollar, but an odd number we lose a dollar. We don't
care whether we get a 2, 4, or 6- only that we get an even number because
we win the same amount regardless. In this way, rolling a 2, 4, or 6 are
related. Formally we can define a relation on A as follows: Given a, b € A,
then a~ b iffa and b are either both even or both odd.
We generalize the previous example in the following definition.
Definition 17.2.5. Given a partition P on A, we may define a binary
relation ~p C AX A as follows: for a, b € A, a ~p biff a and b are both
contained in the same subset in the partition.
A
![Exercise 17.2.6. In the following parts we will considering partitions of R
and the associated binary relations defined by Definition 17.2.5.
(a) Let P = {R₁, R2} where R₁ = {x | x € R, x ≥ 0} and R₂ = {x | x €
R, x < 0}.
(i) Draw the real number line from 5 to 5, and indicate the sets
R₁, R₂ EP (you may indicate the two sets by circling them sepa-
rately).
(ii) Graph the associated binary relation ~p. You only need to graph
from 5 to 5. (Recall that the graph of a binary relation is a set
in the Cartesian plane, as in Figure 17.1.1.)
=
(b) Let P = {..., R_2, R1, Ro, R₁, R₂,...} where Rn
n} for any integer n.
2
{x | x € R, [x] =
(i) Draw the real number line from 5 to 5, and indicate the visible
sets in P.
(ii) Graph the associated binary relation ~p. You only need to graph
from 5 to 5.
—
(c) Let P = {E, O} where E = {x | x € R, [r] is even} and 0 = {r | r €
R, [x] is odd}.
(i) Draw the real number line from 5 to 5, and indicate the sets
E, O EP (a good way to do this is to color the intervals belonging
to E and O with different colors).
(ii) Graph the associated binary relation ~p. You only need to graph
from -5 to 5.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F892e817a-9b32-4eeb-b8fc-5dd7ffde6479%2F09ff7c9b-f9fb-465b-8268-7f630dd58aa8%2Fnsr3vkv_processed.png&w=3840&q=75)
Transcribed Image Text:Exercise 17.2.6. In the following parts we will considering partitions of R
and the associated binary relations defined by Definition 17.2.5.
(a) Let P = {R₁, R2} where R₁ = {x | x € R, x ≥ 0} and R₂ = {x | x €
R, x < 0}.
(i) Draw the real number line from 5 to 5, and indicate the sets
R₁, R₂ EP (you may indicate the two sets by circling them sepa-
rately).
(ii) Graph the associated binary relation ~p. You only need to graph
from 5 to 5. (Recall that the graph of a binary relation is a set
in the Cartesian plane, as in Figure 17.1.1.)
=
(b) Let P = {..., R_2, R1, Ro, R₁, R₂,...} where Rn
n} for any integer n.
2
{x | x € R, [x] =
(i) Draw the real number line from 5 to 5, and indicate the visible
sets in P.
(ii) Graph the associated binary relation ~p. You only need to graph
from 5 to 5.
—
(c) Let P = {E, O} where E = {x | x € R, [r] is even} and 0 = {r | r €
R, [x] is odd}.
(i) Draw the real number line from 5 to 5, and indicate the sets
E, O EP (a good way to do this is to color the intervals belonging
to E and O with different colors).
(ii) Graph the associated binary relation ~p. You only need to graph
from -5 to 5.
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