D and E questions 10. Prove that for subsets X, Y, and Z of a given universal set U(a) X-Y=X-(XY)= (XUY) -Y(b) (X-Y)-Z = X - (YUZ)(c) X-(Y-Z) = (X-Y)U(Xnz)(d) (XUY) - Z= (X-Z) U (Y - Z)(e) X-(YUZ) =(x-Y)^(X-Z)Illustrate each case on a Venn diagram.

Answered: Image /qna-images/answer/1aa43fa8-536d-4265-bf16-febc56ef30c3.jpg

Answered: (d) (XUY) -Z=(X-Z)U(Y-Z) (e) X(YUZ) =…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.7: Relations

Problem 27E: Prove Theorem 1.40: If is an equivalence relation on the nonempty set , then the distinct...

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In Exercises 1324, prove the statements concerning the relation on the set Z of all integers. If 0xy, then x2y2.
Let be a relation defined on the set of all integers by if and only if sum of and is odd. Decide whether or not is an equivalence relation. Justify your decision.
Let x and y be in Z, not both zero, then x2+y2Z+.
10. Prove that for subsets X, Y, and Z of a given universal set U (a) X-Y=X-(XY)= (XUY) -Y (b) (X-Y)-Z=X - (YUZ) (c) X-(Y-Z) = (X-Y)U(Xnz) (d) (XUY) -Z=(X-Z) U(Y - Z) (e) X-(YUZ) =(x-Y)^(X-Z) Illustrate each case on a Venn diagram.
Let C denote the set of all ordered pairs (a, b) with a,b & R. L.e., C:= {(a,b): a, b = R}. Define addition + and multiplication of such pairs by (u, v) + (x, y) = (u + x, v+y) and (u, v) • (x, y) = (ux — vy, uy + vx) R. Together they form a triple . for all u, v, r, y (a) Show that multiplication is associative in . (b) Show that every element (a, b) € C has a negative, and every element (a, b) € C# has an inverse. (c) Prove or disprove: The system of real numbers R is isomorphic to the system . Here, 0 R is the zero of R. (d) True or false? Justify your answer: The triple C, +, > must contain a subfield isomorphic to R.
There are 16 binary relations on the set {0, 1}: (a) { } (d) {(1,0)} (g) {(0, 0), (1, 0)} (j) {(0, 1), (1, 1)} (m) {(0,0), (0, 1), (1, 1)} (b) {(0, 0)} (e) {(1, 1)} (h) {(0, 0), (1, 1)} (k) {(1, 0), (1, 1)} (c) {(0, 1)} (f) {(0, 0), (0, 1)} (i) {(0, 1), (1,0)} (1) {(0,0), (0, 1), (1, 0)} (0) {(0, 1), (1, 0), (1, 1)} (n) {(0,0), (1,0), (1, 1)} (p) {(0, 0), (0, 1), (1, 0), (1, 1)} For a-f use the letters for each relation above. a. List the reflexive relations on the set {0, 1}. b. List the irreflexive relations on the set {0, 1}. c. List the symmetric relations on the set {0, 1}. d. List the transitive relations on the set {0, 1}. e. List the antisymmetric relations on the set {0, 1}. f. List the asymmetric relations on the set {0, 1}.
1. Find all regular points of the set {x = R¹ | {x € R¹ | +3(x1+x2+x3)² 2(x₁+x2+x3)³ +3(x1+x2 + x3)² = 1, x²+x²+x² + x₁ = 1}. 2(x1+x2+x3)³
6. Show that R - Q and R - Z are not homeomorphic.
Let the sets A, B and C be fuzzy sets defined on real numbers by the membership functions: HA(x)= x/(x+1), Ha(x)= 1/(x²+10), µc(x)= 1/10*. Determine mathematical membership functions and graphs of each of the following: 1. AUB X' 2. BnC 3. AUBUC 4. ANBNC 5. An Complement(C) 6. Complement(B) U C 7. Complement(A O B) 8. Complement(A) U Complement(B).
Let the sets A, B and C be fuzzy sets defined on real numbers by the membership functions: HA(x)= x/(x+1), Ha(x)= 1/(x²+10), µc(x)= 1/10*. Determine mathematical membership functions and graphs of each of the following: 1. AUB X' 2. BnC 3. AUBUC 4. ANBNC 5. An Complement(C) 6. Complement(B) U C 7. Complement(A O B) 8. Complement(A) U Complement(B).
6. Let 2 be a set and f, g: → R. Assume max{f(x) + g(x) | x = }, max{f(x) |x€N}, and max{g(x) | x = N} exist. (a) Show max{f(x) + g(x) | x € N} ≤ max{f(x) | x € N} + max{g(x) | x € N}. (b) Show we can have strict inequality in (a).
Let the domain of x and y be all people. Let W(x): "x is a woman". M(y): "y is a man", and L(x.y): "y loves x". Then vxay. (W(x)AM(y))L(xr, y)] means Every man has a woman that loves him. Every man is loved by all women. None of these Every woman is loved by all men. Every woman has a man that loves her.

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(d) (XUY) -Z=(X-Z)U(Y-Z) (e) X(YUZ) = (X-Y)^(X-Z)