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Recall the definition of even numbers. Definition. An integer n is even if n = 2k for some integer k. Define a relation R on Z by x Ry if x + y is even. (a) Show that R is an equivalence relation. Since x + x = 2x is even, x R x, so R is --Select--. Since x + y = y + x+z=2k₁-y + 2k₂ y = 2(k₁+k₂-y) is even, so x R z. Therefore R is ---Select--- (b) Describe the equivalence classes formed by this relation. , x+y will be even whenever y + x is, so R is ---Select--. Suppose that x Ry and y Rz. Then x + y = 2k₁ and y + z = 2k₂ for some k₁, k₂ EZ. Therefore

Recall the definition of even numbers. Definition. An integer n is even if n = 2k for some integer k. Define a relation R on Z by x Ry if x + y is even. (a) Show that R is an equivalence relation. Since x + x = 2x is even, x R x, so R is --Select--. Since x + y = y + x+z=2k₁-y + 2k₂ y = 2(k₁+k₂-y) is even, so x R z. Therefore R is ---Select--- (b) Describe the equivalence classes formed by this relation. , x+y will be even whenever y + x is, so R is ---Select--. Suppose that x Ry and y Rz. Then x + y = 2k₁ and y + z = 2k₂ for some k₁, k₂ EZ. Therefore

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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Recall the definition of even numbers 

Recall the definition of even numbers. Definition. An integer n is even if n = 2k for some integer k. Define a relation R on Z by x Ry if x + y is even. (a) Show that R is an equivalence relation. Since x + x = 2x is even, x R x, so R is ---Select--- . Since x + y = y + x + z = 2k₁ -y + 2k₂ y = 2(k₁ + k₂y) is even, so x R z. Therefore R is ---Select--- (b) Describe the equivalence classes formed by this relation. , x+y will be even whenever y + x is, so R is ---Select--- Suppose that x Ry and y Rz. Then x + y = 2k₁ and y + z = 2k₂ for some k₁, k₂ € Z. Therefore
Transcribed Image Text:Recall the definition of even numbers. Definition. An integer n is even if n = 2k for some integer k. Define a relation R on Z by x Ry if x + y is even. (a) Show that R is an equivalence relation. Since x + x = 2x is even, x R x, so R is ---Select--- . Since x + y = y + x + z = 2k₁ -y + 2k₂ y = 2(k₁ + k₂y) is even, so x R z. Therefore R is ---Select--- (b) Describe the equivalence classes formed by this relation. , x+y will be even whenever y + x is, so R is ---Select--- Suppose that x Ry and y Rz. Then x + y = 2k₁ and y + z = 2k₂ for some k₁, k₂ € Z. Therefore
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