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Problem 11.16. Let X = {XE Ζ+ : x-100): that is, X is the set of all integers from l to 100. For each Y E 9(X) we defin...
Theorem 7.3.5 Let P be a partition of a nonempty set X. Define a relation~on X for all a, b X by ...
Theorem 7.3.5 Let P be a partition of a nonempty set X. Define a relation~on X for all a, b X by defining: Then is an equivalence relation on X. Furthermore, the equivalence classes ofare exactly the elements of the partition P: that is, X/ ~= P. Proof: See page 164 in your textbook. a,b,c,d,e,f partition P = {{a, c, e), {b, f}, {d)) 5 Let A = Give a complete listing of the ordered pairs in the equivalence relation...
Problem 11.21. For k є Z, we define Ak-{x є Z : x-51+ k for some 1 є z} (a) Prove that {Ak : k Z} partitions Z. (b) We...
Problem 11.21. For k є Z, we define Ak-{x є Z : x-51+ k for some 1 є z} (a) Prove that {Ak : k Z} partitions Z. (b) We denote by ~ the equivalence relation on Z that is obtained from the par- tition of part (a). Give as simple a description ofas possible; that is, given condition "C(x,y)" on x and y s x~y if and only if "C(x, y)" holds.
Problem 11.21. For k є Z, we...
Problem 2. Let f be a self-map on a set X. For x,y e X define...
Problem 2. Let f be a self-map on a set X. For x,y e X define x ~ y if and only if f"(x) = f(y) for some integers n, m > 0. Show that ~ is an equivalence relation.
2. Let f : A ! B. DeÖne a relation R on A by xRy i§ f (x) = f (y). a. Prove that R is an equivalence relation on A. b. Let Ex = fy 2 A : xRyg be the equivalence class of x 2 A. DeÖne E = fEx : x 2 Ag...
2. Let f : A ! B. DeÖne a relation R on A by xRy i§ f (x) = f
(y). a. Prove that R is an equivalence relation on A. b. Let Ex =
fy 2 A : xRyg be the equivalence class of x 2 A. DeÖne E = fEx : x
2 Ag to be the collection of all equivalence classes. Prove that
the function g : A ! E deÖned by g (x) = Ex is...
2. Let S 11,2,3,4,5, 6, 7,8,91 and let T 12,4,6,8. Let R be the relation on P (S) detined by for ...
2. Let S 11,2,3,4,5, 6, 7,8,91 and let T 12,4,6,8. Let R be the relation on P (S) detined by for all X, Y E P (s), (X, Y) E R if and only if IX-T] = IY-T]. (a) Prove that R is an equivalence relation. (b) How many equivalence classes are there? Explain. (c) How mauy elements of [ø], the equivalence class of ø, are there? Explain (d) How many elements of [f1,2,3, 4)], the equivalence class of (1,2,3,...
probelms 9.1 9 Modular arithmetic Definition 9.1 Let S be a set. A relation R =...
probelms 9.1
9 Modular arithmetic Definition 9.1 Let S be a set. A relation R = R(,y) on S is a statement about pairs (x,y) of elements of S. For r,y ES, I is related to y notation: Ry) if R(x,y) is true. A relation Ris: Reflexive if for any I ES, R. Symmetric if for any ry ES, Ry implies y Rr. Transitive if for any r.y.ES, Ry and yRimply R. An equivalence relation is a reflexive, symmetric and...
5. Let Zli_ {a + bi l a,b E Z. i2--1} be the Gaussian integers. Define a function for all a bi E Zi]. We call N the norm (a) Prove that N is multiplicative. This is, prove that for all a bi, c+di E Z...
5. Let Zli_ {a + bi l a,b E Z. i2--1} be the Gaussian integers. Define a function for all a bi E Zi]. We call N the norm (a) Prove that N is multiplicative. This is, prove that for all a bi, c+di E Z[i] (b) Prove that if a + r є z[i] is a unit of Zli], then Ma + bi)-1. (c) Find all of the units in Zli
5. Let Zli_ {a + bi l a,b...
3. (a) (5 points) On the set A= R\{0}, let x ~ y if and only...
3. (a) (5 points) On the set A= R\{0}, let x ~ y if and only if x · y > 0. Is this relation an equivalence relation? Prove your answer. (b) (5 points) Let B = {1, 2, 3, 4, 5} and C = {1,3}. On the set of subsets of B, let D ~ E if and only if DAC = EnC. Is this relation an equivalence relation? Prove your answer.
Let T be the relation defined on R given by T = {(x,y)|X, Y E RAx-yeZ}....
Let T be the relation defined on R given by T = {(x,y)|X, Y E RAx-yeZ}. a. Prove T is an equivalence relation. b. Prove Ō =Z c. Find 1.5
Please do problem 9 and write a detailed proof when doing (a) 9. Letbe the relation on the set of non-zero real numbers defined as follows: for r, y E R [0), x~ylf and only if-EQ (a) Prove thatis an...
Please do problem 9 and write a detailed proof when doing
(a)
9. Letbe the relation on the set of non-zero real numbers defined as follows: for r, y E R [0), x~ylf and only if-EQ (a) Prove thatis an equivalence relation. (b) Determine the equivalence class of π.
9. Letbe the relation on the set of non-zero real numbers defined as follows: for r, y E R [0), x~ylf and only if-EQ (a) Prove thatis an equivalence relation. (b)...
Theorem 7.3.5 Let P be a partition of a nonempty set X. Define a relation~on X for all a, b X by defining: Then is an equivalence relation on X. Furthermore, the equivalence classes ofare exactly the elements of the partition P: that is, X/ ~= P. Proof: See page 164 in your textbook. a,b,c,d,e,f partition P = {{a, c, e), {b, f}, {d)) 5 Let A = Give a complete listing of the ordered pairs in the equivalence relation...
Problem 11.21. For k є Z, we define Ak-{x є Z : x-51+ k for some 1 є z} (a) Prove that {Ak : k Z} partitions Z. (b) We denote by ~ the equivalence relation on Z that is obtained from the par- tition of part (a). Give as simple a description ofas possible; that is, given condition "C(x,y)" on x and y s x~y if and only if "C(x, y)" holds.
Problem 11.21. For k є Z, we...
Problem 2. Let f be a self-map on a set X. For x,y e X define x ~ y if and only if f"(x) = f(y) for some integers n, m > 0. Show that ~ is an equivalence relation.
2. Let f : A ! B. DeÖne a relation R on A by xRy i§ f (x) = f
(y). a. Prove that R is an equivalence relation on A. b. Let Ex =
fy 2 A : xRyg be the equivalence class of x 2 A. DeÖne E = fEx : x
2 Ag to be the collection of all equivalence classes. Prove that
the function g : A ! E deÖned by g (x) = Ex is...
2. Let S 11,2,3,4,5, 6, 7,8,91 and let T 12,4,6,8. Let R be the relation on P (S) detined by for all X, Y E P (s), (X, Y) E R if and only if IX-T] = IY-T]. (a) Prove that R is an equivalence relation. (b) How many equivalence classes are there? Explain. (c) How mauy elements of [ø], the equivalence class of ø, are there? Explain (d) How many elements of [f1,2,3, 4)], the equivalence class of (1,2,3,...
probelms 9.1
9 Modular arithmetic Definition 9.1 Let S be a set. A relation R = R(,y) on S is a statement about pairs (x,y) of elements of S. For r,y ES, I is related to y notation: Ry) if R(x,y) is true. A relation Ris: Reflexive if for any I ES, R. Symmetric if for any ry ES, Ry implies y Rr. Transitive if for any r.y.ES, Ry and yRimply R. An equivalence relation is a reflexive, symmetric and...
5. Let Zli_ {a + bi l a,b E Z. i2--1} be the Gaussian integers. Define a function for all a bi E Zi]. We call N the norm (a) Prove that N is multiplicative. This is, prove that for all a bi, c+di E Z[i] (b) Prove that if a + r є z[i] is a unit of Zli], then Ma + bi)-1. (c) Find all of the units in Zli
5. Let Zli_ {a + bi l a,b...
3. (a) (5 points) On the set A= R\{0}, let x ~ y if and only if x · y > 0. Is this relation an equivalence relation? Prove your answer. (b) (5 points) Let B = {1, 2, 3, 4, 5} and C = {1,3}. On the set of subsets of B, let D ~ E if and only if DAC = EnC. Is this relation an equivalence relation? Prove your answer.
Let T be the relation defined on R given by T = {(x,y)|X, Y E RAx-yeZ}. a. Prove T is an equivalence relation. b. Prove Ō =Z c. Find 1.5
Please do problem 9 and write a detailed proof when doing
(a)
9. Letbe the relation on the set of non-zero real numbers defined as follows: for r, y E R [0), x~ylf and only if-EQ (a) Prove thatis an equivalence relation. (b) Determine the equivalence class of π.
9. Letbe the relation on the set of non-zero real numbers defined as follows: for r, y E R [0), x~ylf and only if-EQ (a) Prove thatis an equivalence relation. (b)...
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