Let T be the set of all triangles in a plane with R a relation in T given byR = {(T1, T2) : T1 is congruent to T2}. Show that R is an equivalence relation.

Q: Let A C B and let S be an equivalence relation on B. Prove that R = {(x,y) E A × A : (x, y) E S} %3D…

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Q: Is the relation R on X is reflexive, symmetric, weakly antisymmetric, antisymmetric, transitive?…

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Q: 4. Consider the following binary relation p on the plane R² (x, y)p(u, v) if y? – a² = v² – u².…

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Q: Let A be the set of all triangles defined by R = {(A1, A2) : A1 =A2). Show that R is an equivalence…

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Q: Let A = {x, y, z}. Write down a relation R on A that is not reflexive, not sym- metric, and not…

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Q: Consider the following relation R on R: (x, y) ER if and only if y E Q. What are the properties of…

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Q: . Define a relation R from R to R as follows: For every (x, y) ERX R, (x, y) E R means that y = r.…

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Q: The relation R defined on a set A is antisymmetric if (a, b) ER = (b, a) E R for (A) Every (a, b) ER…

A: A relation R is said to be antisymmetric if (a,b)∈R and (b,a)∈R implies a=b

Q: Let R be the relation defined on P({1,., 100}) by ARB if and only if |AU B| is even. Is R reflexive?…

A: Given:  R is a relation defined on P 1, 2, … , 100 by A R B  if and only if A ∪ B is even. We have…

Q: Let A={a,b,c,d,e}. Find a relation R on A which is symmetric and transitive but not reflexive

A: Reflexive relation means every term of a set is related with itself

Q: Let R be the relation on N defined by R = {(n, m) EN x N: m=n+1). Then R is surjective. True False

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Q: Let A = Z. Define R to be the relation where a-b if ab > 0. Then R is equivalence %3D relation.

A: A relation R on a non empty set A is said to be equivalence relation if R is reflexive, symmetric…

Q: The relation R is defined on R^2 by (x1, y1) ∼ (x2, y2) if and only if x1 − x2 ∈ Z,for (x1, y1) and…

A: a. The relation R is defined on R^2 by (x1, y1) ~ (x2, y2) if and only if x1 − x2 ∈ Z,for (x1, y1)…

Q: Let R be a relation on R × R such that (,71)R¢xI2) if and only if xıyı = x2y2. -) Is (2.3)R(1,-6) a…

A: It is given that R be a relation on ℝ×ℝ such that Rx2,y2x1,y1 iff x1y1=x2y2 For part (a) According…

Q: Consider a relation R on Z defined as follows: xRy → xy is even. a. Give a counterexample to show…

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Q: Let - be the relation on R\{0} given by x ~ y if and only if xy is positive. Prove that this is an…

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Q: Let L be the set of all lines in XY plane and R be the relation in L defined asR = {(L1, L2) : L1 is…

A: Line L is parallel to itself. This means L is parallel to L.So, (L,L)∈RSo, R is reflexive.

Q: Let R be a relation defined on P(Z) defined by (A, B) E R if and only if AN B + Ø. Prove or disprove…

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Q: Show that the relation R defined in the set A of all triangles as R = {(T1, T2) :T1is similar to…

A: Given,relation R defined in the set A of all triangles as R = {(T1, T2) :T1 is similar to T2}.

Q: Define a relation C from R to R as follows: For any(x, y) ∈ R × R, (x, y) ∈ means that x2 + y2 = 4…

A: Topic = function

Q: Define a map t: R × R → R × R by t(a, b) = (a+ b, a - b). Prove that t is a one-to-one…

A: The solution is given as

Q: Let T be a topological construct on R defined by the form T= {Q₂R} U² E₂ = (a, ∞o) a € R} Find…

A: Let T be a topological construct onℝ defined by the formT={ϕ,ℝ}∪Ea=(a,∞);a∈ℝ

Q: In the Cartesian plane = {(x, y): x,y E R} consisting of points with rectangular coordinates (x, y),…

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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: (b) Let - be the relation on C defined by z ~ w if and only if z + w e R. Proye that is an…

A: This can be solved as follows:

Q: Let L be the set of all lines in a plane and R be the relation in L defined asR = {(L1, L2) : L1 is…

A: The given relation R is defined as follows.

Q: Let R be the relation on R defined by R= {(x, y) E R × R | x² + y² = 4 or x = y²}. Answer each of…

A: Given that R be the relation on ℝ defined by,R=x,y∈ℝ×ℝx2+y2=4 or x=y2

Q: Consider the relation R on R defined by (x, y) E R if and only if |æ – y| = ||x| – ly||- This…

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Q: Let X = R. Define an equivalence relation ~ on X by x ~ y if %3D and only if x – y is an integer.…

A: Let X, ?X be a topological space, and let ~ be an equivalence relation on X. The quotient set,…

Q: Let R be a relation on Z. Then Ris an equivalence relation if it is defined by xRy if and only if…

A: R is an equivalent relation iff R is reflexive. R is symmetric. R is transitive.

Q: Define a relation C from R to R as follows: For any (x, y)  £ R x R.  (x, y)  C means that x² + y²…

A: Given: Define a relation C from R to R as follows,for any (x,y)∈R×R (x,y) C means that x2+y2=4

Q: Let A = {1, 2, 3, 4} and R a relation on A whose matri: 1 0 1 0 0 1 0 1 is Mr = 0 0 1 1 0 0 1…

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Q: Consider the relation R defined on Z given by a Ry 6|(x + 5y). a) Prove that R is an equivalence…

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Q: Consider the relation S on N defined by uSy⟺5|(y+u). Is it reflexive?

A: No it is not a reflexive relation

Q: Define a relation C from R to R as follows: For any (x, y) ∈ R x R, (x, y) ∈ C means that x² + y² =…

A: (x,y) ∈ R × R (x,y) ∈ C means    x2 + y2 =4

Q: Let : [0, 1] → [0.c], te- and let w be a 1-form on R. What and foe, and why? is the relation between…

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Q: 4. Let R be the relation xRy if 3| (x – y) on Z. Prove that R is an equivalence relation.

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Q: Let R be the ternary relation on Z that is defined as R:= {(x,y, z) E Z × Z × Z : a² + y² + 2² < 5}…

A: If a relation is defined on Z, then the variables in the relation can have only integer values to…

Q: Let S be the following relation on ℂ: S = {(x, y) ∈ ℂ2 : y − x is real}. Prove that S is an…

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Q: Let R be the relation defined on P({1,., 100}) by ARB if and only if |AU B| is even. s R reflexive?…

A: We will check one by one what properties R satisfies and which not.

Q: 8. Let A = Z* U {0}, define relation R on A × Z* by (k,l)R(m,n) → kn = lm. (a). Prove that R is an…

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Q: A relation R is defined on Z by xRy if x*y >= 0. Prove or disprove the following: a. R is…

A: Let us consider the given relation R on set X. R is said to be reflexive for any x in X, x R x. R…

Q: RUR and RNR symmetric relations on X. are Show that R=Z xZ is an equivalence relation.

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Q: Show that the relation R in R defined as R = {(a, b) : a ≤ b}, is reflexive andtransitive but not…

A: Given, the relation R in R defined as R = {(a, b) : a ≤ b}.

Q: Let A = Z+ x Z+. Define a relation R on A as follows: For all (r, y) and (2, w) in A, (r, y)R(z, w)…

A: Given that: A=Z+xZ+. Relation is:        For all (x, y) and (z, w) in A, (x, y)R(z, w)⇔x+w=z+y…

Q: 36. For any two points (a, b) and (c, d) of the plane, define (a, b) ~ (c, d) provided that a? + b?…

A: Given : Any two points (a, b) and (c, d) of the plane. Definition :  a, b≃ c, d provided that…

Q: - Use the connectivity relation Mg- MR V M V M to find the transitive 2. closure of the following…

A: To find- Use the connectivity relation MR* = MR ∨ MR2 ∨ MR3 to find the transitive closure of the…

Q: Let R be the relation defined on P({1,.…, 100}) by ... ARB if and only if |AUB| is even. Is R…

A: Relation: - A subset of the productA× B could be a relation R from a non-empty set A to a non-empty…

Q: Define a relation R from R to R as follows: For all (x, y) ER×R, (x, y) eR if, and only if, x = y? +…

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Q: Let R be the relation on Z defined by x Ry if and only if x+ 3y is even. Prove that R is an…

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