(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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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 \( a, b \in A \), then \( a \sim b \) iff \( a \) 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 \( \sim_P \subset A \times A \) as follows: for \( a, b \in A \), \( a \sim_P b \) iff \( a \) and \( b \) are both contained in the same subset in the partition. \( \triangle \)

Transcribed Image Text:**Exercise 17.2.6**: In the following parts, we will consider partitions of the real numbers \(\mathbb{R}\) and the associated binary relations defined by Definition 17.2.5.
(a) Let \(P = \{R_1, R_2\}\) where \(R_1 = \{x \mid x \in \mathbb{R}, x \geq 0\}\) and \(R_2 = \{x \mid x \in \mathbb{R}, x < 0\}\).
(i) Draw the real number line from -5 to 5, and indicate the sets \(R_1, R_2 \in P\) (you may indicate the two sets by circling them separately).
(ii) Graph the associated binary relation \(\sim_P\). 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 \(P = \{\ldots, R_{-2}, R_{-1}, R_0, R_1, R_2, \ldots\}\) where \(R_n = \{x \mid x \in \mathbb{R}, \lfloor x \rfloor = n\}\) for any integer \(n\).
(i) Draw the real number line from -5 to 5, and indicate the visible sets in \(P\).
(ii) Graph the associated binary relation \(\sim_P\). You only need to graph from -5 to 5.
(c) Let \(P = \{E, O\}\) where \(E = \{x \mid x \in \mathbb{R}, \lfloor x \rfloor \text{ is even}\}\) and \(O = \{x \mid x \in \mathbb{R}, \lfloor x \rfloor \text{ is odd}\}\).
(i) Draw the real number line from -5 to 5, and indicate the sets \(E, O \in P\) (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 \(\
Expert Solution

This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 4 images

Recommended textbooks for you

Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated

Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education

Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY

Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated

Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education

Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY

Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,

