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(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
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please do Exercise 17.2.6 part B and C

 

**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:**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. \( \triangle \)
**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 \(\
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 \(\
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