1. Given a relation R on a set A, prove that if R is transitive, then so is R-

Q: Given a relation R on a set A, prove that if R is transitive, then so is R-1.

A: Suppose A be any set and R is relation defined on A. Then R is transitive if whenever (a,b)∈R and…

Q: Given a non empty set X, consider P(X) which is the set of all subsets of X.      Define the…

A: The relation is defined as follows.

Q: Prove that for any set relation R (we did not pick it to be "on" any particular set) R* =N{Q:RC Q^ Q…

A: Given that any set relation R.

Q: Given the relation R = {(u,t),(v,t),(w,t),(w,u),(w,v)} on a set A = {t,u,v,w}, what is/are the…

A: topic - relation

Q: 1. Consider the set Z in which the relation R is defined by aRb if and only if a + 3b is divisible…

A: According to question given that Set Z in which the relation R is defined by aRbIf and only if a+3b…

Q: Prove the understated statements concerning the relation < on the set Z of all integers . If    x…

A: x = y  and     O < z

Q: A relationR on a set A is called equivalence if R is:

A: Equivalence Relation: A relation R on a set A is said to be an equivalence relation if and only if…

Q: 4. Suppose R is a relation on a set A. The complementary relation R is defined by letting a R b if a…

A:

Q: Assume R is a relation Dver set A. a) prove Ris anti-ceflective 1 Diaw graph G(LE) where the…

A: A relation R on a set X is said to be anti-reflective if no element  of X is related  to itself…

Q: Let A be a set and R be a symmetric and transitive relation on A. Prove the following statement: If…

A:

Q: A map f:x-y is said to be an open map if for every open set U of X, the set f(u) is open in Y. Show…

A: Given, a map f: X→Y is said to be an open map if for every open set U of X, the set f(U) is open in…

Q: Let R be a transitive relation on set A. Define Q to be {(r Prove or disprove that (RUR®)n (QU Qº)…

A:

Q: We defined the relation between sets by A ~ B means that there there is a 1-1 correspondence ƒ : A →…

A: We define the relation ~ between sets by A~B means that there is a one-one correspondence f:A→B Show…

Q: -prove that for any bounded Set E, There is abounded set A That is a countable intersection fopen…

A: The solution is:

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

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Q: Use a membership table to prove the following relation: * (An B) U (A n Bº) = A

A: We have to use membership table.

Q: for an equivalence relation R on A, both sets, the class of equivalence classes of R —A/R— is a set.

A:

Q: 1. Prove that “divides" is a transitive relation. That is, prove that, for all integers a, b, c, if…

A: I the given question we have to prove that , for all integers a,b,c, if a|b and b|c, then a|c.

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

A:

Q: Without using the ?∞i=1 Ri representation of R+, Prove that for any set relation R (we did not pick…

A: Given: Let us consider the given, R+=∩Q:R⊆Q∧Q is transitive and prove that for any set relation R.

Q: 4. Suppose Rand S are symmetric relations on a set A. Prove that RoS is symmetric iff Ro S = So R.…

A: Suppose R∘S is symmetric Let s, r∈S∘R ⇔r, s∈R∘S⇔s, r∈R∘S ( since R∘S is symmetric)  So, R∘S=S∘R

Q: In each item below, verify that ~ is an equivalence relation on the given set. (a) Let S = Z x N.…

A:

Q: 4. Let X be any finite set and define the relation R on P(X) by R= {(E,F)||E| = |F\, E, F € P(X)}.

A:

Q: 3. Given a relation R on a set A, prove that if R is transitive, then so is R-1.

A: We are given  that R is a transitive relation on a set A  Transitive Relation  A relation on a set X…

Q: Let R be the relation on set {a,b,c,d} containing ordered pairs (a,c),(b,b),(b,a)(c,a)(c,d) and…

A: Here given that the set is A = {a, b, c, d} and the relation on A is given by R = {(a, c), (b, b),…

Q: Suppose A is a set and R is a reflexive and transitive relation on A. Show that the

A:

Q: Let A be a set and let ∼ be an equivalence relation on A. Let x, y ∈ A. Prove that [x] = [y] if and…

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Q: Let S be the set of integers. If a, b E S, define aRb if ab >= 0. Is R anequivalence relation on…

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Q: Let S and T be finite sets, and let f: S T be a map. Then |S| = IS {}| YET Select one: O True OFalse

A:

Q: A relation R, is defined on Z, the set of integers by aRb + a - b is even. Show that R is…

A:

Q: Show that the derived set E′ of any set E is closed.

A: Here is the proof .

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: Problem 17. (a) Prove that <r is a reflexive, transitive relation on P(E*). (b) Prove that =r is an…

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Q: Let X be the set of all subsets of ℤ . Define a relation R on X as follows: Given two subsets of…

A:

Q: Let R be a relation defined on the set Z by aRb if floor value of a = floor value of b. What…

A:

Q: See the instructions for Exercise (19) on page T00 from Section 3,1 (a) Proposition. Let R be a…

A: We use definition for following question

Q: Let A be a set and R be a reflexive relation on A. Prove that R is an equivalence relation if and…

A:

Q: Consider the set Q = Z ×(Z 10), and the relation tupon QDefined as: (a, b) ↑ (c, d) e a·d=b·c a - d…

A:

Q: Consider the set Z in which the relation R is defined by aRb if and only if a + 3b is divisible by…

A: Since you have asked multiple question, we will solve the first question for you. If youwant any…

Q: Let R be a relation from set A to B, if R is symmetric then its inverse will be...... Answer (Please…

A: Given R is a relation from set A to B. Let R is symmetric relation and R-1 be the inverse relation…

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

A:

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: Prove that a set A is connected iff A is and interval.

A:

Q: 1. Consider the following relations defined on the set {a, b, c}. For each relation, determine…

A: (a) Reflexive functions are the ones whose domain and range are the same  for instance a,a∈R1 So…

Q: (1) Let A-(a,b,c,d), define a relation R on set A as follows: R-((a,a). (a,b), (b,a), (b,b), (c,c),…

A: Reflexive relation:  A relation R on set X is said to be reflexive if for all elements x∈X we have…

Q: For each of the following relations on sets {a, b, c, d}, decide whether it is reflexive, symmetric,…

A: As per Bartleby guidelines i can answer only one question or its sub three parts kindly upload rest…

Q: Let S be a set with a finite number of elements, and let f: S - S be a map _ (a) If fis onto, can If…

A: We are given that S is a set having finite number of elements and f : S--&gt;S be a map.Before we go…

Q: 5. Without using the U R representation of R*, Prove that for any set relation R (we did not pick it…

A: We need to Prove that for any set relation R (we did not pick it to be "on" any particular set R+ =…

Family of Curves

A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.

XZ Plane

In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.  

Euclidean Geometry

Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.

Lines and Angles

In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.