A relation R is defined on R by a R b if a – b e Z. Prove that R is an equivalence relation and determine the equivalence classes [1/2] and [v2].
Q: Suppose that R is a reflexive and transitive relation on a set A. Define a new relation E on A by…
A:
Q: A relation < is defined on R? by (x1, x2) < (y1, y2) if and only if x1 < yi and x1 + x2 2 yı + y2.…
A:
Q: Let R be a relation on Z defined by (x,y)ER if and only if 5(x-y)=0. Formall state what it means for…
A:
Q: 1. Show that which of these relations on the set of all functions on Z→Z are equivalence relations?…
A:
Q: Let f: X → Y and define x~y if f (x) = f (y). Show that - is an equivalence relation on X.
A: The objective is to show that ~ is an equivalence relation on X.
Q: Let R be a relation on the set A of ordered pairs of positive integersdefined by (x, y) R (u, v) if…
A: Given:
Q: Let a relation ∼ on R2 = {(a, b): a, b ∈ R} be defined as: (a, b) ∼ (c, d) if and only if | a | + |…
A:
Q: b. Define a relation SER XR by S ={ (x,y) E R × R |x – y € Z }. Prove that S is an equivalence…
A: By equivalence relation, we mean that the relation is reflexive, symmetric, and transitive.Here we…
Q: Consider the relation R on Z defined by the rule that (a, b) E R if and only if a + 26 is even.…
A:
Q: On sets, the relation x R y given by there exists a bijective function from the set x to the set y'…
A:
Q: In the Cartesian plane = {(x, y): x,y E R} consisting of points with rectangular coordinates (x, y),…
A:
Q: Consider the relation R on R2 given by for any (a,b),(c,d)∈R 2, (a,b)R(c,d) if and only if a 2 +b…
A: According to the given information, Consider a relation R on R2 as:
Q: a Define a relation R on N by (a,b) e Rif and only if EN Which of the following properties does R…
A:
Q: Q7. Let R be a relation on the set Z defined by the following rule: for all a, b e Z, a Rb if and…
A:
Q: 1. Show that which of these relations on the set of all functions on Z-→Z are equivalence relations?…
A:
Q: ) Let A = {1,2,3, 4} × {1,2, 3, 4}, and define a relation R on A by (1, Yı)R(x2, Y2) if r1 + y1 = x2…
A: Equivalence relation means it should satisfy reflexive, symmetric, Transitive conditions.…
Q: Consider the binary relation R defined on (N* x N*) by: V (X1. Yı). (x2. Y2) E N* x N*, (x1. Yı) R…
A: Order relation (definition)-: A relation R is called an order relation if it is reflexive,…
Q: Let ≥ be a relation defined on sets A and B by A ≥ B iff there exists a surjective function f : A →…
A:
Q: Let D be an integral domain and define a relation - on DxD such that b 0 and d=0 by (a,b) - (c,d) if…
A: Equivalence Relation Definition : A relation R on a set A is said to be an equivalence relation if…
Q: Consider the relation R defined on Z given by a Ry 6|(x + 5y). a) Prove that R is an equivalence…
A:
Q: A relation R is called atransitive if aRb and bRc implies cRa. Show that R is reflexive and…
A: Given a relation R is called atransitive if aRb and bRc⇒cRa Now shows in the following that R is…
Q: Let p be an integral domain and define a relation - on DxD such that bz0 and dz0 by (a,b) -~(c,d) if…
A: R is the symbol for relation. Symmetric: if (a,b)R(c,d) ∈ D×D ⇒ad=bc ⇒bc=ad ⇒(c,d) R…
Q: Show that the relation R on Z defined by aRb if and only if 5a−3b is even, for a, b ∈ Z, is an…
A: Let R be a relation from a set A to itself. R is said to be an equivalence relation if 1) R is…
Q: Verify that the relation "... is tangent to... at a" is an equivalence relation on the functions…
A: An equivalence relation on a set S is a relation ~ on S such that: 1) x~x for all x∈S then the…
Q: 4. For r, y E R, let z y if and only if (x- y) E Q. Show that defined as such is an equivale…
A: ~ is defined as : For all, x, y in R, x~y if and only if (x-y) belongs to Q Reflexivity : x~x for…
Q: A relation R on R² is defined by ((x1, y1), (2, 2)) R if Prove that R is an equivalence relation.…
A:
Q: . The relation R on Z defined by a Rb if a? = b² (mod 4) is known to be an equivalence relation.…
A: Consider the given information.
Q: Prove whether the following statements are true or false: b) The relation R defined by xRy7[(x-y)…
A:
Q: Prove that the relation is an equivalence relation. 1. For x,y E R say x is congruent to y modulo Z…
A:
Q: Problem 17. (a) Prove that <r is a reflexive, transitive relation on P(E*). (b) Prove that =r is an…
A:
Q: 1. A relation R is defined on Z* × Z* by (m,n)R(p, q) → m+q = n + p . (a). Prove that R is an…
A:
Q: Let p be an integral domain and define a relation - on D×D such that b=0 only if ad = bc: Prove that…
A: As per the question, we need to solve (8). Let D be an integral domain. The relation is defined as…
Q: Define a relation T on R by xT y if and only if (sin^2) x + (cos^2) y = 1. (a) Prove that T is an…
A:
Q: Consider the relation on Z×(Z\ {0}) defined by (m,n)R(m’,n’) provided that mn’ = m’n. Prove that…
A:
Q: 4|(x+3y)
A:
Q: Let A = {-5, -4, −2, 0, 3, 6, 8), and define an equivalence relation R on A as follows: (x, y) E R…
A: Let A = {-5, -4, -2, 0, 3, 6, 8}, and define an equivalence relation R on A as follows: (x, y) in R…
Q: . Let R be the relation on N defined by xR y if x and y share a common factor other than 1.…
A:
Q: Show that which of these relations on the set of all functions on Z-→Z are equivalence relations?…
A:
Q: Let R be a relation from A to B and S a relation from B to A. Prove or disprove that if S of R =…
A:
Q: 2. Let H ≤ G and define = on G by a = b iff a¯¹b € H. Show that =µ is an equivalence relation.
A:
Q: Let p be an integral domain and define a relation ~ on Dx D such that b 0 and d 0 by (a,b) ~ (c, d)…
A:
Q: In this exercise, you want to show that a relation RC A? is transitive + R" C R for all n EN=…
A:
Q: Define a relation T on R as follows: for all x and y in R, x T y if and only if x2=y2. (a) Prove…
A: Given that a relation T on R such that for all x and y in R, xTy if and only if x2=y2 (a) We have to…
Q: Let S be the following relation on C10). S=[(x,y) E (C\[0]): y/x is real). Prove that S is an…
A:
Q: 5. Let R be a relation defined on Z by a Rb if and only if 3 | (a + 2b). (a) Prove that R is an…
A: A relation to be an equivalence relation must satisfy the following three properties: 1. Reflexive…
Q: On P2, define a relation p(x) ~ q(x) if p(0) = q(0). (a) Show that this is an equivalence relation…
A:
Q: Which of the following statements is TRUE? sgn( (5 8) )=1 Let R be a relation on Z defined by xRy if…
A: ?Relation R is not Reflexive, because 2+2=4 does not divisible by 5. So , (2,2) not in R. ?…
Q: 40
A: By using the definition of equivalence relation solution is given as follows :
Q: For a,y E R, let z~y if and only if (x - y) € Q. Show that ~ defined as such is an equivalence…
A:
Q: Let R be a relation and S its reverse. Show that R is injective if and only if S is well defined,…
A: Consider the provided question, Given that, R be a relation and S its reverse. Let R is a relation…
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
- For any relation on the nonempty set, the inverse of is the relation defined by if and only if . Prove the following statements. is symmetric if and only if . is antisymmetric if and only if is a subset of . is asymmetric if and only if .Label each of the following statements as either true or false. Let R be a relation on a nonempty set A that is symmetric and transitive. Since R is symmetric xRy implies yRx. Since R is transitive xRy and yRx implies xRx. Hence R is alsoreflexive and thus an equivalence relation on A.10. Let and be mappings from to. Prove that if is invertible, then is onto and is one-to-one.
- Exercises 33. Prove Theorem : Let be a permutation on with . The relation defined on by if and only if for some is an equivalence relation on .Let R be the relation defined on the set of integers by aRb if and only if ab. Prove or disprove that R is an equivalence relation.A relation R on a nonempty set A is called asymmetric if, for x and y in A, xRy implies yRx. Which of the relations in Exercise 2 areasymmetric? In each of the following parts, a relation R is defined on the set of all integers. Determine in each case whether or not R is reflexive, symmetric, or transitive. Justify your answers. a. xRy if and only if x=2y. b. xRy if and only if x=y. c. xRy if and only if y=xk for some k in . d. xRy if and only if xy. e. xRy if and only if xy. f. xRy if and only if x=|y|. g. xRy if and only if |x||y+1|. h. xRy if and only if xy i. xRy if and only if xy j. xRy if and only if |xy|=1. k. xRy if and only if |xy|1.
- 5. For each of the following mappings, determine whether the mapping is onto and whether it is one-to-one. Justify all negative answers. (Compare these results with the corresponding parts of Exercise 4.) a. b. c. d. e. f.For each of the following parts, give an example of a mapping from E to E that satisfies the given conditions. a. one-to-one and onto b. one-to-one and not onto c. onto and not one-to-one d. not one-to-one and not ontoComplete the proof of Theorem 5.30 by providing the following statements, where and are arbitrary elements of and ordered integral domain. If and, then. One and only one of the following statements is true: . Theorem 5.30 Properties of Suppose that is an ordered integral domain. The relation has the following properties, whereand are arbitrary elements of. If then. If and then. If and then. One and only one of the following statements is true: .
- For each of the following mappings f:ZZ, determine whether the mapping is onto and whether it is one-to-one. Justify all negative answers. a. f(x)=2x b. f(x)=3x c. f(x)=x+3 d. f(x)=x3 e. f(x)=|x| f. f(x)=x|x| g. f(x)={xifxiseven2x1ifxisodd h. f(x)={xifxisevenx1ifxisodd i. f(x)={xifxisevenx12ifxisodd j. f(x)={x1ifxiseven2xifxisoddLet a and b be constant integers with a0, and let the mapping f:ZZ be defined by f(x)=ax+b. Prove that f is one-to-one. Prove that f is onto if and only if a=1 or a=1.27. Let , where and are nonempty. Prove that has the property that for every subset of if and only if is one-to-one. (Compare with Exercise 15 b.). 15. b. For the mapping , show that if , then .