need help with proving discrete math HW, please try write clearly and i will give a thumb up thanks!!
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need help with proving discrete math HW, please try write clearly and i will give a thumb up thanks!! Let A and be B be...
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...
please help with discrete math HW. please write clearly and dont make up an answer.............i will...
please help with discrete math HW. please write clearly and dont
make up an answer.............i will rate for the best answer thank
you. will appreciate if you can answer all 5 parts.
(L,) complete lattice and f L >L an order-preserving function (i) Let a, bE L with a b. Define a, b]= {z € L | a b} Show that (a, b], 3) is a complete lattice (ii) Consider X = {x € L \ f(x) = x}. The...
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,...
need help with discrete math HW, please try write clearly and i will give a thumb up thanks!! (i) Prove that every comp...
need help with discrete math HW, please try write clearly and i
will give a thumb up thanks!!
(i) Prove that every complete lattice has a unique maximal element. (ii) Give an example of an infinite chain complete poset with no unique maximal 1 element (iii) Prove that any closed interval on R ([a, b) with the usual order (<) is a complete lattice (you may assume the properties of R that you assume in Calculus class) (iv) Say that...
[12] 5. Let A = {1, 2, 3, 4, ..., 271}. Define the relation R on...
[12] 5. Let A = {1, 2, 3, 4, ..., 271}. Define the relation R on A x A by: for any (a,b), (c,d) E AXA, (a,b) R (c,d) if and only if a +b=c+d. (a) Prove that R is an equivalence relation on AX A. (b) List all the elements of [(3,3)], the equivalence class of (3, 3). (c) How many equivalence classes does R have? Explain. (d) Is there an equivalence class that has exactly 271 elements? Explain.
Please answer all!! 17. (a) Let R be the relation on Z be defined by a...
Please answer all!!
17. (a) Let R be the relation on Z be defined by a R b if a² + 1 = 62 + 1 for a, b e Z. Show that R is an equivalence relation. (b) Find these equivalence classes: [0], [2], and [7]. 8. Let A, B, C and D be sets. Prove that (A x B) U (C x D) C (AUC) Ⓡ (BUD).
A. (Leftovers from the Proof of the Pigeonhole Principle). As before, let A and B be finite sets with A! 〉 BI 〉 0 and let f : A → B be any function Given a A. let C-A-Va) and let D-B-{ f(a)} PaRT...
A. (Leftovers from the Proof of the Pigeonhole Principle). As before, let A and B be finite sets with A! 〉 BI 〉 0 and let f : A → B be any function Given a A. let C-A-Va) and let D-B-{ f(a)} PaRT A1. Define g: C -> D by f(x)-g(x). Briefly, if g is not injective, then explain why f is not injective either. Let j : B → { 1, 2, 3, . . . , BI}...
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...
Exercise 3 (Cantor-Bernstein-Schröder). Let f: A → B and g: B → A be injective maps....
Exercise 3 (Cantor-Bernstein-Schröder). Let f: A → B and g: B → A be injective maps. We define recursively the sets C = UCn Co = A \ g(B), Cn+1 = g(f(Cn)), nƐN and a new map h: A → B by if x E C, f (x) h(x) = if x 4 C, g='(x) where the preimage g¬1(x) is well-defined since g is injective and x E g(B) in that case (check that!). Show that h is bijective. Conclude...
Let X, Y be two nonempty sets and let f : X → Y. For a,...
Let X, Y be two nonempty sets and let f : X → Y. For a, b X we write a ~ b iff f(a) = f(b). Prove that~is an equivalence relation on X Write lely for the equivalence class of x e X with respect to “~" Express [ely in terms of the function f: Irl, = {re x : f(z') a: b: ?? J. (I d o not want to see ..|x ' = {x"e X : r,...
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...
please help with discrete math HW. please write clearly and dont
make up an answer.............i will rate for the best answer thank
you. will appreciate if you can answer all 5 parts.
(L,) complete lattice and f L >L an order-preserving function (i) Let a, bE L with a b. Define a, b]= {z € L | a b} Show that (a, b], 3) is a complete lattice (ii) Consider X = {x € L \ f(x) = x}. The...
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,...
need help with discrete math HW, please try write clearly and i
will give a thumb up thanks!!
(i) Prove that every complete lattice has a unique maximal element. (ii) Give an example of an infinite chain complete poset with no unique maximal 1 element (iii) Prove that any closed interval on R ([a, b) with the usual order (<) is a complete lattice (you may assume the properties of R that you assume in Calculus class) (iv) Say that...
[12] 5. Let A = {1, 2, 3, 4, ..., 271}. Define the relation R on A x A by: for any (a,b), (c,d) E AXA, (a,b) R (c,d) if and only if a +b=c+d. (a) Prove that R is an equivalence relation on AX A. (b) List all the elements of [(3,3)], the equivalence class of (3, 3). (c) How many equivalence classes does R have? Explain. (d) Is there an equivalence class that has exactly 271 elements? Explain.
Please answer all!!
17. (a) Let R be the relation on Z be defined by a R b if a² + 1 = 62 + 1 for a, b e Z. Show that R is an equivalence relation. (b) Find these equivalence classes: [0], [2], and [7]. 8. Let A, B, C and D be sets. Prove that (A x B) U (C x D) C (AUC) Ⓡ (BUD).
A. (Leftovers from the Proof of the Pigeonhole Principle). As before, let A and B be finite sets with A! 〉 BI 〉 0 and let f : A → B be any function Given a A. let C-A-Va) and let D-B-{ f(a)} PaRT A1. Define g: C -> D by f(x)-g(x). Briefly, if g is not injective, then explain why f is not injective either. Let j : B → { 1, 2, 3, . . . , BI}...
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...
Exercise 3 (Cantor-Bernstein-Schröder). Let f: A → B and g: B → A be injective maps. We define recursively the sets C = UCn Co = A \ g(B), Cn+1 = g(f(Cn)), nƐN and a new map h: A → B by if x E C, f (x) h(x) = if x 4 C, g='(x) where the preimage g¬1(x) is well-defined since g is injective and x E g(B) in that case (check that!). Show that h is bijective. Conclude...
Let X, Y be two nonempty sets and let f : X → Y. For a, b X we write a ~ b iff f(a) = f(b). Prove that~is an equivalence relation on X Write lely for the equivalence class of x e X with respect to “~" Express [ely in terms of the function f: Irl, = {re x : f(z') a: b: ?? J. (I d o not want to see ..|x ' = {x"e X : r,...
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