Suppose R1 is a relation on domain X1 and R2 is a relation on domain X2 Define a new relation R whose domain is X1×X2: (x1,x2)R(x1′,x2′) if x1R1x1′ and x2R2x2′. (a) Show that if R1 and R2 are both partial orders, then R is also a partial order.   (b) Now change the relation R so that (x1,x2)R(x1′,x2′) if x1R1x1′ OR x2R2x2′. Give an example in which R1 and R2 are partial orders but R is not a partial order.

Q: 2. For each of the following relations, indicate whether it is a partial order, a strict order, or…

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Q: Let A = Z+ x Z+. Define a relation R on A as follows: For all (x, y) and (z, w) in A, (x, y)R(z, w)…

A: We will find out the required values.

Q: Find, for each part of the question, an example of a set S and two relations R1 C S x S and R2 C S x…

A: Since the questions contains many subparts. As per the guidelines I am answering the first three…

Q: 7. Let (a, b), (c,d) E R2. Define a relation ~on R² by (a, b) (c,d) if 2a - b = 2c-d. (a) Show that…

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Q: Consider the relation R on R, where R = {(a, b) | ab ≥ 0} . (a) Is the relation R reflexive,…

A: is a relation on

Q: R.3. Consider the relation R = {(m,n):m,n E N & m|n}. Prove that R defines a partial order on the…

A: A relation R is said to be partial order on a set X if the relation is reflexive, transitive and…

Q: (i) Let A = {-1, 1,2,3,4}, X = {(-1, 1), (1, 1), (2, 2), (3, 3), (4, 4), (1, −1), (1, 4), (4, 1)}.…

A: We are allowed to do a maximum three sub parts as per bartleby guidelines

Q: 2. Let R = {(x, y) = R x R = |x] + [y] =4}. Then R is a relation on R. Is R an equivalence relation…

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Q: Suppose that f : A → B is a function, viewed as a relation (set of ordered pairs). Let f* denote the…

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Q: (i) Give an example of a relation S on A = {1,2, 3, 4} with S is reflexive, not symmetric and not…

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Q: 3. Show that the relation R (called lexicographic ordering) defined on Z x Z by: (a, b)R(c, d) if (a…

A: The lexicographic ordering relation R on ℤ×ℤ is defined by, a, bRc, d if a>c or a=c and b≥d Show…

Q: consider the relation R on R, where R = {(a, b) | a ≤ b+1}. (a) Is the relation R reflexive,…

A: i.Let  then So the relation  holds for all .So  is reflexive.ii.Since the relation is reflexive so…

Q: What is the point of a partial order? a. We would like to generalize the properties of the relation…

A: Numbers are easily comparable. But two elements may satisfy either of the relations, a=b, a<b,…

Q: 4. Define a relation R on N by aRb if a and b have no common factors except 1. (a) Give an example…

A: We have define  a relation R on natural numbers (N) by aRb if a and b have no common  factors except…

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 R be the relation defined on Z by aRb if a + b is even. I have the following equivalence classes…

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Q: R.5. Define a relation < on the set of ordered pairs of real numbers (x, y) as follows: (X:V1)< (x2,…

A: The relation ≺ is defined as follows:  x1,y1≺x2,y2 if and only if x1&lt;x2 or x1=x2 and y1≤y2

Q: Consider the relation R on R given by R = {(a, b) | a−b > 3 or a−b = 0}. (a) Show that R is not an…

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