The relation R is defined on R^2 by (x1, y1) ∼ (x2, y2) if and only if x1 − x2 ∈ Z,for (x1, y1) and (x2, y2) in R^2 a) Prove that R is an equivalence relation. Be sure to use the correct format, including labelling b) ] Find the equivalence class [(2, 3)] and sketch it in the x-y plane.

Q: Consider the collection of sets {An: n E R}, where for each n E R An := {(x, y) E R²: 2x + n}. y =…

A:

Q: Define relation R on (Z+ U {0}) × Z+ by (a, b)R(c,d) if ad = bc. Describe the equivalence class [(2,…

A: Our objective is given below:

Q: Let ∼be the relation on Z defined as follows: For every m, n ∈Z, m ∼n if, and only if, 5 | (m2 −n2…

A: (a) For every m ∈ Z, 5 | (m2 −m2) = 0      Hence, m ~ m for all m ∈ Z      Hence, ~ is reflexive.

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

A: Given: R is the relation defined on P1,....,100 by A R B.           A∩B is even.We need to check…

Q: Assume that you know that ~ is an equivalence relation on R\ {0} defined as follows: xy > 0. Find…

A:

Q: ~ Define an equivalence relation on Z by a b if a - b is even. Show that . equivalence relation.…

A:

Q: 6. Are the following relations on R equivalence relations? Justify your answer! a) R = {(x, y): x -…

A:

Q: - b if and only if f (a) = f(b). Show that ~ is an Let f: A → B be a surjection. Define a ~…

A: Prove ~ is an equivalence relation. 1) Reflexive Since f is a function, f(a) = f(a) and so a ~ a for…

Q: Please help me out

A: Equivalence Classes: The set S is divided into equivalence classes by ~, which is an equivalence…

Q: efine the relation ~ on the set N of positive integers by a~b if and only if  a = b(10^k) for some…

A:

Q: Prove relation B is an equivalence relation

A:

Q: 4. Describe all Consider the relation R on R x R defined by (x1, y1) R(x2, y2) iff x₁ - 2y2 = x₂ -…

A:

Q: For every element r in A, there is an element y in B such that (x, y) is in F. Can you please…

A: Relation: The relation from a set A to a set B is a subset of Cartesian product A×B. Function: The…

Q: Exercise 1.2. Determine whether the following relations ~ is an equivalence relation: (1) nm in Z if…

A: Step 1:Step 2:Step 3: Step 4:

Q: 5. Let~ be an equivalence relation on a set S. We define f: S→S/~ f(x) = [a], where a € S such that…

A:

Q: Does the follorin relations hip define y as a real furction of x Why," AN

A:

Q: Prove that these relations on the set of all functions from Z to Z are equiva- lence relations.…

A:

Q: How many distinct equivalence classes exist in the relation R defined as below: x Ry + 3| (2x - )

A: