Please do Exercise 17.3.4 part A and please show step by step and explain Example 17.3.1. Define a binary relation ~on R by ay iff x² = y².Then is an equivalence relation.PROOF. We wish to show that is reflexive, symmetric, and transitive.17.3 EXAMPLES OF EQUIVALENCE RELATIONS595(reflexive) Given x R, we have r² = x², so r~x.(symmetric) Given x, y R, such that ~ y, we have x² = y². Sinceequality is symmetric, this implies y² = x², so y ~ r.(transitive) Given x, y, z € R, such that zy and y~z, we have ² = y²and y² = 2². Therefore x² = 2², since equality is transitive. Hence ~ z. Exercise 17.3.3. Show that each of these binary relations is an equivalencerelation.(a) The binary relation ~ on R defined by r~y iff r² - 3x = y² − 3y.(b) The binary relation ~ on R defined by r ~ y iff x − y ≤ Z. (*Hint*)(c) The binary relation ~ on Nx N defined by (a1, b₁) ~ (a2, b₂) iff a1b₂ =a2b1. (*Hint*)(d) The binary relation ~ on C defined by 21 ~ 22 iff |21| = |22|-Equivalence relations are often defined in terms of functions. For in-stance, Example 17.3.1 involves the function f: R→ R defined by f(x) =2², and xy if and only if f(x) = f(y). Similarly, Exercise 17.3.3 involvesthe function g: R → R defined by f(x) = x² – 3x, and ~ y if and only ifg(x) = g(y). Both of these cases follow the following pattern:Given a function f : A → B, define a binary relation on A by: a₁ ~ a2 iff f(a₁) = f(a₂).Other examples that we've seen also follow this same pattern:Exercise 17.3.4. Following the pattern that we've shown for Example 17.3.1and Exercise 17.3.3, define the following equivalence relations in terms offunctions.(a) Exercise 17.3.3 part (d)(b) Exercise 17.3.3 part (e)(c) The binary relation in Example 17.3.2.

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Exercise 17.3.3. Show that each of these binary relations is an equivalence relation. (a) The binary relation~ on R defined by r ~y iff r² - 3x = y² − 3y. (b) The binary relation ~ on R defined by r ~ y iff r − y € Z. (*Hint*) (c) The binary relation ~ on Nx N defined by (a1, b₁) ~ (a2, b2) iff a1b2 = a2b1. (*Hint*) (d) The binary relation on C defined by 21 ~ 22 iff |21| = |22|- Equivalence relations are often defined in terms of functions. For in- stance, Example 17.3.1 involves the function f: R→ R defined by f(x) = 2², and ~ y if and only if f(x) = f(y). Similarly, Exercise 17.3.3 involves the function g: R → R defined by f(x)=x²-3x, and r ~ y if and only if g(x) = g(y). Both of these cases follow the following pattern: Given a function f : A → B, define a binary relation on A by: a₁ ~ az iff f(a₁) Other examples that we've seen also follow this same pattern: Exercise 17.3.4. Following the pattern that we've shown for Example 17.3.1 and Exercise 17.3.3, define the following equivalence relations in terms of functions. (a) Exercise 17.3.3 part (d)

Exercise 17.3.3. Show that each of these binary relations is an equivalence relation. (a) The binary relation~ on R defined by r ~y iff r² - 3x = y² − 3y. (b) The binary relation ~ on R defined by r ~ y iff r − y € Z. (Hint) (c) The binary relation ~ on Nx N defined by (a1, b₁) ~ (a2, b2) iff a1b2 = a2b1. (Hint) (d) The binary relation on C defined by 21 ~ 22 iff |21| = |22|- Equivalence relations are often defined in terms of functions. For in- stance, Example 17.3.1 involves the function f: R→ R defined by f(x) = 2², and ~ y if and only if f(x) = f(y). Similarly, Exercise 17.3.3 involves the function g: R → R defined by f(x)=x²-3x, and r ~ y if and only if g(x) = g(y). Both of these cases follow the following pattern: Given a function f : A → B, define a binary relation on A by: a₁ ~ az iff f(a₁) Other examples that we've seen also follow this same pattern: Exercise 17.3.4. Following the pattern that we've shown for Example 17.3.1 and Exercise 17.3.3, define the following equivalence relations in terms of functions. (a) Exercise 17.3.3 part (d)

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.7: Relations

Problem 11E: Let be a relation defined on the set of all integers by if and only if sum of and is odd. Decide...

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