2. Let ƒ : A → B be a function, and ~ an equivalence relation on B. For a₁, a2 € A, we say a₁ ≈ a2 if ƒ(a₁) ~ f(a₂). Prove that ≈ is an equivalence relation.

Q: 4. Let f : X → Y define a relation on the set X by „Ry if and only if (x) = fy). Is f an equivalence…

A: f is an equivalence relation if f satisfies three properties. 1. if f is a reflexive relation  2. if…

Q: Recall that a |b is a relation on the natural numbers meaning “a divides b.” It is defined by a |b…

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Q: Let S={1,2,3} and T={-2,-1,0}. Define a relation R from S to T as follows: For all (x,y) in SxT,…

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Q: 3. Let f: A → B and g: B → C be functions. Recall that the composition of g and f is the function go…

A: a) To prove that if g and f are injective, then go f is injective, we need to show that for any two…

Q: Let RC RX R with {(x, y) |x² =5 y²}. Characterize R in terms of whether it is reflexive,…

A: A reflexive, symmetric, transitive binary relation on a set is an equivalence relation. In other…

Q: Let A and B be two nonempty sets, f: A → B be a function and R be an equivalence relation on the set…

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Q: R.2. Consider the relation R=- ={x,y):x.yeR&]x=}<3}. Is R an equivalence relation on set of real…

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Q: Let G = {-2, 0, 2} and H = {4, 6, 8} and define a relation V from G to H as follows: For every (х,…

A: (a)To verify:

Q: You are given two graphs G and H with V(G) = {a, b, c, d} and V(H) = {1, 2, 3}. 1. How many…

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Q: Find a set A and a relation R such that the relation is symmetric, and not reflexive and transitive.

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Q: 2. Let R be the relation on the set A = (1, 2, 3, 4, 5, 6, 7} defined by the rule (a,b) e R if the…

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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: Let Y = {0,1,2} and Z = {0,1} and define a relation R from Z to Y as follows: Given {x, y} € Z x Y,…

A: Relation : A relation between two sets is a collection of ordered pairs containing one object from…

Q: 2. Let R₁ be a relation on the set A and R₂ be a relation on the set B. Let R be the relation…

A: A relation is a subset of Cartesian product of two sets. Therefore, every element in a relation set…

Q: IF A = { 1,1,2,3,5} B = {2,4} A relation S from A to B such that (x, y) ∈ S means that x ≤ 3 and y…

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Q: R = cN0.8, where R is the number of miles of roadway in a city and N is the population.†   (a)…

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Q: 9. Let R+ denote the set of positive real numbers. Define a relation E on R+ as follows: a Eb if and…

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Q: A function F from a set A to a set B is a relation with domain A and co-domain B that satisfies the…

A: Relation: Let A  and B  be two sets then relation between A nd B means  the collection of the…

Q: Let S {(a,b): a,be R, a 0}. Show that the relation R defined on S by (a, b) R(c,d) if = ad be is an…

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Q: Suppose that f : A → B and g : B → C are both injective functions, prove that the function go f : A…

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Q: 1. Let R be the relation on N defined by "aRb if EN or 5| (b – a)". So for example (6,2) € R and…

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Q: Let f : A B be function. Prove that the relation a - b if and only if f(a) = f(b) is an equivalence…

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Q: Let A = {2, 3, 4, 5} and B = {6, 8, 10}. A relation R is defined from A to B by a R b if a | b and…

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Q: QUESTION 2 Define a relation on Nx N as follows: (a, b) R (c,d) a+d=b+c (2.1) Is (3,6) R (3,5)?…

A: We have to show that the given relation is an equivalence relation.

Q: Let f: A → B be a function, and let C be any set such that BCC. True or False: f : A → C is always a…

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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.

A:

Q: 5. Let A = {6, 8, 10, 15, 17} and B = {2, 3, 4, 5} and define a relation R: A → B as follows: For…

A: Given: A={6, 8, 10, 15, 17} B={2, 3, 4, 5} Relation R:A→B is: For all x,y∈A×B∈R if and only if…

Q: Let f : X → Y be a function. Define a relation R in X given byR = {(a, b): f(a) = f(b)}. Examine…

A: Given:

Q: 4. Let the relation R = { (a, b) e Z × Z l(a + b) = 1 (mod 5) }. a. Is the relation transitive?…

A: A relation R between two sets A and B is the set of all ordered pairs a,b such that a∈A and b∈B. We…

Q: 2. There is a function f : X →Y . Let's define a relation R in a set X, by a formula: (xRy) → (f(x)…

A: Given f: X → Y and R is the relation in the set X such that x R y ⇔ f(x) = f(y)

Q: 3. Let S = R × R and R be the relation on S defined by (a, b)R(c, d) iff a+c=b+d. Determine whether…

A: Reflexive: (x,y) R (x,y) As x+y=x+y Symmetric : if (a,b)R(c,d) then a+b=c+d. Since c+d=a+b then…

Q: Consider the relation R = {(1,1);(2,3);(3,5);(4,7)}. a. Prove that R is a function from {1,2,3,4}…

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Q: Let R be a relation from A to B and S a relation from B to A. Prove or disprove that if S of R =…

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Q: Determine whether the following relation is an equivalence relation or not. (a) G = {(x, y) |æ…

A: We have to determine whether the given relation is equivalence relation or not.

Q: Define an equivalence relation on R² by: ¤1, Y1) ~ (x2, Y2) e if and only if (*1, Yı) and (x2, Y2)…

A: The complete solutions are given below