5. Construct a smallest binary relation S defined on the set {w,r, y, z} such that S satisfies all of the following requirements: • S contains (r, y). • S is not transitive. • Sis not antisymmetric. • S is not symmetric. |S| is even.

Q: d) Let X = {1,2,3, 4}. Define a binary relation R on subsets of X by ARB if and only if A and B have…

A:

Q: ). Let R be binary relation on N defined by xRy if and only if x- 2s ys x+2. Is R reflexive? Is R…

A: We will usethe definition of reflexive, symmetric, antisymmetric and transitive realtion to answer…

Q: 2) Let R be binary relation on N defined by r Ry if and only if r <y< 2r. Is R reflexive? Is R…

A:

Q: Suppose that A and B are bounded, nonempty subsets of R. Prove that AU B is bounded and that sup(AU…

A: Consider the provided question, Suppose A and B are bounded, nonempty subsets of R.

Q: 8. Draw the directed graph of the binary relation S on B = { a, b, c, d, e} by S = {(a, b),(b,…

A:

Q: (2) Show that if f has support in a compact set A CR and g has support in a compact set B C R then f…

A:

Q: Let P be a set on which a binary relation < is defined such that, for all r, y, z € P, (a) * <x is…

A: Consider the provided question, Since, A relation R that is reflexive, anti-symmetric and transitive…

Q: In a Cofinite topology of X the collection of subsets a) r = {AcX|A ±Ø or A is finite} b) r = {Ac X|…

A: Let X be a non-empty set, then the collection of subsets of X whose complements are finite along…

Q: Draw the directed graph of the relation R on A = {1,2,3,4,5,6,7,8} defined as R = {(a,b)|a,b e A and…

A: Here, the relation is: R=(a,b)|a,b∈A and a≡b mod 3 That is: a-b is divisible by 3. So, 7-1=6 is…

Q: 3. Let R be the relation defined on Z by aRb if and only if 5 | (4a + b). Prove that R is an…

A: We have given relation on ℤ and defined by  aRb if and only if 5|(4a+b). We can  see here for any…

Q: PROBLEM 30. For (x, y) and (u, v) in R2, define (x, y) ~ (u, v) if x² + y² = u? + v?. Prove that…

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Q: Suppose R and S are symmetric relations on a set A. Prove that RoS is symmetric iff RoS = So R.

A:

Q: the book is "First Course on Fuzzy Theory and Applications"

A: To prove: All the α–cuts of any fuzzy set A defined on Rn­ are convex if and only if

Q: 1. Determine whether or not the relation R on the set of all people is reflexive, symmetric,…

A:

Q: Let D be the binary relation defined below. For all u, w e 2, u D w → w=u +7 or w= u - 7. Show…

A: Solution of given problem is as below: A binary relation is equivalance relation if it satisfies the…

Q: 1. Prove or Disprove: For any set A, there exists a relation R on A such that R is both symmetric…

A:

Q: Let P be a partition of the nonempty set A. For x, y E A, define xQy if and only if there exists CEP…

A:

Q: If M and N are two norms on space X then prove that M and N are equivalent if and only if a subset…

A: Any two norms M and N on a norm space is called equivalent if  M≤N≤M. in the given question we have…

Q: 5. • S contains (w, r) and (2, u). • S is not antisymmetric. • S is not symmetric. • is not…

A: The binary relation based on given set {u,v,w,x,y,z}

Q: 1. Determine whether the relation R on the set of all people is reflexive, symmetric, antisymmetric,…

A: solution

Q: 7. Let X be a set with |X| = n € Z4. Prove that for any set Y, |Y| = n if and only if there exists a…

A:

Q: QUESTION 4 Define a relation R on Z+, such that (Z+, R) is a poset that has infinite number of…

A:

Q: 9. E is the binary relation defined on Z as follows: For all m, n E Z, m Enm-n is even. Is this an…

A: Equivalence relation

Q: on K³defined by: (a, b, c) ~ (d, e, f) if and only if ak e K – {0} such that (a, b, c) = k(c, e, d)…

A:

Q: 3. Prove that the image of a measurable set E under the translation 4 is measurable and m(E+a)…

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: 1. Suppose R and S are symmetric relations on a set A. Prove that Ros is symmetric if Ros=SOR

A: Given:  To prove R and S are symmetric relations.

Q: Let Z be with the co-finite topology If A = {1,3,5, 7,.. }, then A A O O O

A: As Z is with co-finite topology. Every open neighborhood of an infinite set must contain infinitely…

Q: Theorem 6.17. Every compact subset C of R contains a maximum in the set C, i.e., there is an m E C…

A: To prove that, every compact subset C of ℝ contains a maximum in the set C, that is, there is an m∈C…

Q: 1 Recall that for x, y ≤ R, |x − y| ≤ Rf is the absolute value of x - y. That is, if x-y≥0 |x - y =…

A: Please see the below picture for detailed solution.

Q: 1. Given the binary relation R on R defined by „R, if and only if x > y – 1, for each of the…

A:

Q: Consider the binary relation R = {(x,y),(x,z),(z,x),(z,y)} on the set (x,y,z}, which one of the…

A: We have to find which one of the options is true for relation R.

Q: Construct a smallest binary relation S defined on the set {w, x, y, z} such that S satisfies all of…

A: Given: S contains (x. y). S is not transitive. S is not antisymmetric. S is not symmetric. ISI is…

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 z be the set of integers and R be the equivalence relation on ZxZ defined by: (a.b)R(c.d) if and…

A: Given that R is a relation on Z×Z defined by (a,b) R(c,d) if and only if a+d=b+c.

Q: Let X = [0, 1]. Consider the following claim: “For every complete binary relation on a non-empty set…

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Q: . Determine whether or not each of the binary relations R defined on the given sets A are reflexive,…

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 A = {2, 4} and B = {1, 3, 5} and define rela- tions U, V, and W from A to B as follows: For…

A: (a) First obtain the 3 relations as follows.

Q: Construct a smallest binary relation S defined on the set {w, x, y, z} such that S satisfies all of…

A: The objective is to determine the smallest binary relation S defined on the set &lt;w,x,y,z&gt; such…

Q: A relationR is called a partially ordered set if R is: .a reflexive .b antisymmetric .C transitive…

A:

Q: Consider X = {a1, a2, ..., an} Can a topology of X with 3 open and one with 4 open be equivalent?

A: We have to verify the given statement.

Q: (c) Let ~ be a relation defined on Z by a R b if and only if a’ = b³ (mod 4) . Determine the…

A:

Q: 4. Define a binary relation R on the set of integers Z by (a, b) E R if and only if |a – b| < 1. Is…

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Q: Let G={(m,n)∣m,n∈N x N and gcd(m,n)=1}, and define the relation q on G according to (a,b) q…

A: A totally ordered set is a set G together with a relation ≤ that satisfies the following conditions.…

Q: 5. Prove or disprove: for any set A, there exists a relation R on A such that R is both symmetric…

A: Symmetric : If aRb , then bRa , for every a , b∈A Antisymmetric : If aRb and bRa , then a=b

Q: Consider the subset G {r• /2: x € Z} of R. Show that G is closed un- der addition.

A: The given subset is: G= x·2 : x∈ℤ

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: Prove the understated statements concerning the relation &lt; on the set Z of all integers .   If O…

A: Prove the understated statements concerning the relation &lt; on the set Z of all integers .   If O…