1. Give and example of: (a) a binary relation that is reflexive, transitive, not symmetric and not antisymmetric. (b) a strict partial order that has exactly three linear extensions, or explain why one cannot exist.

Q: 9.3. Let A = {0,1}. Determine all the relations on A.

A: The objective is to determine all the relations on A.

Q: Prove, or find a counterexample to, the following conjectures: (a) If R and S are partial orderings…

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Q: 5. For each of the following partial orders, draw a Hasse diagram, find the maximal and minimal…

A: (a) Given, R is a partial order on the set                0, 2, 4, 6, 8, 10 such that (x, y)∈R if…

Q: Every equivalence relation is reflexive. True False

A: A relation is said to be an equivalence relation if and only if the relation R is reflexive,…

Q: Theorem 3.2.14. (i) The union of a finite collection of closed sets is closed.

A: we have to show that ; The union of a finite collection of closed set is closed.

Q: Let A=[n].  What are the maximium and minimum possible size of an equivalence relation on A?  Prove…

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Q: Is the relation with the directed graphs shown below an equivalence relation? 4. 6.

A: Let  A be a nonempty set. A relation ~ on the set A is an equivalence relation provided that ~ is…

Q: Prove that every well order is a linear order,

A: A set  is said to be well-ordered if it has a total order and has the property that every non-empty…

Q: {(a, b) acb, a and b are finite set of integers}

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Q: Show that any 2-cut relation (for > 0) of a fuzzy equivalence relation results in a crisp…

A: Let's take the fuzzy relation: R = 10.800.10.20.810.400.900.41000.10010.50.20.900.51 Fuzzy tolerance…

Q: Suppose R is both an equivalence and partial order relation. Show that every equivalence class of R…

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Q: Let A= {1,2,3,4,5} and define a binary relation Ramong the subsets of A as follows: XRY XCY. a) Is R…

A: A relation 'R' is defined on set 'A' is said to be: Reflexive : ∀a∈A      aRa Symmetric: if aRb…

Q: 1. Suppose that the matrix representation of a relation is given by 11 1 1 0 1 1 1 0 0 11 0 0 0 1 MR…

A: A zero-one matrix is the matrix representation of a relation. The zero-one matrix is the matrix Mij…

Q: The domain for the relation R is Z × Z. (a,b) R (c,d) if and only if a <c and b< d. Indicate whether…

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Q: D. aRb means 'a – b = 3k' where KEZ %3D - 3. the requirements for an equivalence relation R on a set…

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Q: Show that the relation R = ∅ on a nonempty set S is symmetric and transitive, but not reflexive

A: See the attachment.

Q: The universal relation A x A on A is an equivalence relation anti-symmetric a partial ordering…

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Q: (4) Let T be the partial order relation on P({1,2,..., 20}) defined by ATB if and only if |A < |B|…

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

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Q: Let R be a relation over a set A = {a,b,c,d}. What kind of order is R?  Note: Select the order that…

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Q: 4 Let (A, 3) be a countable dense linear order without endpoints. Prove that there exists an…

A: Solution: Given that (A,≼) is a countable dense linear order without end points. Since A is…

Q: What are the minimal elements of the partial order? What are the maximal elements of the partial…

A: The minimal element is an element which is not greater than any other element Therefor I, A ,F are…

Q: Suppose R1 is a relation on domain X1 and R2 is a relation on domain X2 Define a new relation R…

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Q: a) For the relation R on the set {1, 2, 3, 4}, decide whether it is reflexive, whether it is an…

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Q: Prove or disprove: If R and S are partial orderings on A, then R ∩ S is also a partial ordering on…

A: Partial order relation of intersection of two partial order set

Q: True or False: A Partial Order Relation is transitive, symmetric, and reflexive. True False

A: A partial order Relation is transitive, antisymmetric and reflexive.

Q: Let 2 be a non-empty set. (a) Show that the inclusion relation in P (2) is an order relation. (b)…

A: Let Ω be a non-empty set. a) To show that the inclusion relation in P(Ω) in an order relation. X,Y ∈…

Q: List elements of S(partial ordering relation).

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Q: [1 1 1] Consider the following zero-one matrix 0 1 1 representing a relation on the ordered set {a,…

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Q: For every nonempty set A, there exists a relation R which is both an equivalence relation and a…

A: A relation R is said to be equivalence relation if R is Reflexive ,Symmetric and Transitive . R is…

Q: The directed looped graph of a partial order relation has non-trivial cycles True False Every…

A: To check the given statements are true or false. (1)The directed looped graph of a partial order…

Q: Is the following statement true or false? If R is any symmetric relation on a set A, then R is…

A: If R is any symmetric relation on a set A, then R-1 is symmetric.Proof :1. Let R be any symmetric…

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: In 1-8, a number of relations are defined on the set A = {0, 1, 2, 3}. For each relation: a. Draw…

A: Given: The relation defined on the set A=0,1,2,3 is R2=0,0,0,1,1,1,1,2,2,2,2,3 We want to determine…

Q: A partial order relation is that which is A Reflexive, symmetric and transitive B Reflexive and…

A: A (non-strict) partial order is a homogeneous binary relation ≤ over a set P satisfying particular…

Q: 1. Let A = {cat,dog,bird,rat} and R be a relation on A defined by {(x,y) : x and y have at least one…

A: We have to solve given problem:

Q: 4- The orthogonal relation is A: symmetric and reflective relation B: symmetric and not reflective…

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Q: Congruent is.. Relation on Z Equivalence. anti reflexive anti symmetric Equivalence partial order

A: Here we choose the correct option.

Q: 1) Either prove or disprove the following statements: i) transitive. If binary relations T and S are…

A: Given: T and S are transitive  Suppose a,b∈T∩S, b,c∈T∩S ⇒a,b∈T, b,c∈T Now as T is transitive . So,…

Q: Determine whether the relation with the directed graph shown is an equivalence relation. Determine…

A: Let R be the given relation shown in the directed graph  A = {a,b,c,d} R be relation on A

Q: (d) Is M antisymmetric? If not why not? (g) If M is an equivalence relation, what are the…

A: Note:- As per our guidelines, we can answer the first three sub-parts of this problem. Please…

Q: 1. Let A = {0,1,2,3} and R a relation over A: R = {(0,0).(0,1).(0,3).(1,1),(1,0).(2,3).(3,3)} Check…

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Q: The system represented by the input-output relation, yln]= 2x[n]is an example for a first order…

A: Solve as below

Q: For any partial order there may be a number of linear orders that are consistent. For example, for…

A: This is a question from Relation.

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: Which of the following is not a property of a partial ordering relation? O Reflexivity Symmetry…

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Q: Determine whether the relations represented by the directedgraphs shown below are…

A: We need to determine whether the relations represented by the directed graphs (given in the…

Q: 2.4 (34). Let R be a partial order relation on a finite set S. Show that S has a maximal element…

A: Given: Let R be a partial order relation on the finite set S

Q: 4. a. Define the following Terms giving one example each: i. Partial Ordering Relations ii.…

A: We will answer the first question as you didn't mention any. Please resubmit the other questions…